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In contrast to graph-based models for complex networks, hypergraphs are more general structures going beyond binary relations of graphs. For graphs, statistics gauging different aspects of their structures have been devised and there is undergoing research for devising them for hypergraphs. Forman-Ricci curvature is a statistics for graphs, which is based on Riemannian geometry, and that stresses the relational character of vertices in a network through the analysis of edges rather than vertices. In spite of the different applications of this curvature, it has not yet been formulated for hypergraphs. Here we devise the Forman-Ricci curvature for directed and undirected hypergraphs, where the curvature for graphs is a particular case. We report its upper and lower bounds and the respective bounds for the graph case. The curvature quantifies the trade-off between hyperedge(arc) size and the degree of participation of hyperedge(arc) vertices in other hyperedges(arcs). We calculated the curvature for two large networks: Wikipedia vote network and Escherichia coli metabolic network. In the first case the curvature is ruled by hyperedge size, while in the second by hyperedge degree. We found that the number of users involved in Wikipedia elections goes hand-in-hand with the participation of experienced users. The curvature values of the metabolic network allowed detecting redundant and bottle neck reactions. It is found that ADP phosphorilation is the metabolic bottle neck reaction but that the reverse reaction is not that central for the metabolism.
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Forman-Ricci Curvature for Hypergraphs
Wilmer Leal,1, 2, Guillermo Restrepo,2, 3 Peter F. Stadler,1, 2, 3, 4, 5, 6 and J¨urgen Jost2, 6
1Bioinformatics Group, Department of Computer Science,
Universit¨at Leipzig, H¨artelstraße 16-18, 04107 Leipzig, Germany
2Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany
3Interdisciplinary Center for Bioinformatics, Universit¨at Leipzig, H¨artelstraße 16-18, 04107 Leipzig, Germany
4Institute for Theoretical Chemistry, University of Vienna, W¨ahringerstraße 17, 1090 Vienna, Austria
5Facultad de Ciencias, Universidad Nacional de Colombia, KR 30-45 3, 111321, Bogot´a, Colombia
6The Santa Fe Institute, 1399 Hyde Park Rd., 87501, Santa Fe, New Mexico, USA
(Dated: November 20, 2018)
In contrast to graph-based models for complex networks, hypergraphs are more general structures
going beyond binary relations of graphs. For graphs, statistics gauging different aspects of their
structures have been devised and there is undergoing research for devising them for hypergraphs.
Forman-Ricci curvature is a statistics for graphs, which is based on Riemannian geometry, and that
stresses the relational character of vertices in a network through the analysis of edges rather than
vertices. In spite of the different applications of this curvature, it has not yet been formulated for
hypergraphs. Here we devise the Forman-Ricci curvature for directed and undirected hypergraphs,
where the curvature for graphs is a particular case. We report its upper and lower bounds and the
respective bounds for the graph case. The curvature quantifies the trade-off between hyperedge(arc)
size and the degree of participation of hyperedge(arc) vertices in other hyperedges(arcs). We calcu-
lated the curvature for two large networks: Wikipedia vote network and Escherichia coli metabolic
network. In the first case the curvature is ruled by hyperedge size, while in the second by hyperedge
degree. We found that the number of users involved in Wikipedia elections goes hand-in-hand with
the participation of experienced users. The curvature values of the metabolic network allowed de-
tecting redundant and bottle neck reactions. It is found that ADP phosphorilation is the metabolic
bottle neck reaction but that the reverse reaction is not that central for the metabolism.
I. INTRODUCTION
Hypergraphs are used to model systems whose objects
have not only binary relationships; instead, interactions
simultaneously involve multiple members [1, 2]. Exam-
ples of these systems are found in physics, biology, chem-
istry, computer science, combinatorial optimization, sci-
entometrics and several other fields [1, 3–8]. Hypergraphs
reduce to (ordinary) graphs when all relationships (hy-
peredges) are binary. Graphs have been widely used as
a mathematical model for different systems and their
mathematical properties have been extensively studied,
which include devising statistics gauging aspects of their
structures, such as vertex degree and its distributions,
clustering coefficients, betweenness centrality and more
recently Forman-Ricci curvature.
As hypergraphs are a generalization of graphs, sev-
eral of the graph statistics have been extended to hy-
pergraphs, e.g. vertex and hyperedge degrees, clustering
coefficients [3, 9] and spectral properties [10]. Most of
the commonly used quantities focus on vertices. As the
crucial structure of a graph is, however, given by the
set of its edges rather than by its vertices, we should
systematically define and evaluate quantities assigned to
the edges rather than to the vertices. In this paper we
develop the Forman-Ricci curvature for hypergraphs (di-
rected and undirected) and calculate it for networks of
wilmer@bioinf.uni-leipzig.de
different sizes and research fields.
II. FORMAN-RICCI CURVATURE OF
EDGES/ARCS IN GRAPHS
Recently various notions of “curvature” have been pro-
posed for graphs and other, more general, discrete struc-
tures and applied to detect various local or global prop-
erties of such structures [2, 11–19]. The name of “cur-
vature” may seem somewhat strange in this context. In
differential, and more abstractly, in Riemannian geome-
try, curvature has been found to encode and express local
and global features of smooth manifolds equipped with
metric tensors [20]. Those features themselves usually do
not depend on an underlying smooth structure, and this
has lead to abstract theories of generalized curvatures on
metric spaces. On graphs, these generalized curvatures
are particularly easy to define and to evaluate. They can
also shed considerable light on other quantities that have
been introduced in network analysis without such a clear
conceptual background as those curvatures. The simplest
among these generalized curvatures is the Ricci curvature
introduced by Forman for simplicial complexes [21]. As
graphs are one-dimensional simplicial complexes, we can
readily evaluate this curvature. As explained in detail in
Section II A, for an edge e={i, j}with vertices i, j with
degrees diand dj(the degree of a vertex is the number
of its neighbors, that is, of those other vertices that are
directly connected to it by an edge), the Forman-Ricci
curvature is simply 4didj. The number 4 serves the
arXiv:1811.07825v1 [cs.DM] 19 Nov 2018
2
purpose of normalization, to make the curvature of cycle
graphs vanish. The minus signs are also conventional, to
align this curvature with the Ricci curvature of Rieman-
nian geometry. Thus, edges connecting vertices of large
degree have very negative curvature values, and the first
step in the analysis of an empirical network might con-
sist in identifying the most negatively curved edges as
the most important ones for the cohesion of the network
or for the canalization and distribution of information or
activity in the network.
Since the definition of the Forman-Ricci curvature of
an edge in an undirected graph is so clear and sim-
ple, it can be readily generalized to, for instance, di-
rected or weighted graphs, and also to structures in which
more than two elements are related. Forman himself
had introduced this curvature notion already for possibly
weighted, simplicial complexes [21]. A simplicial com-
plex is characterized by the requirement that whenever a
collection of kelements stands in relation, then this also
holds for any subcollection. This leads to mathematically
very nice properties, and simplicial complexes are basic
structures in algebraic topology, but for the modelling
and analysis of empirical data sets, we may want to re-
lax or perhaps even completely abandon that condition.
That leads us to hypergraphs, which are collections of
vertices (undirected hypergraphs) or collections endowed
with direction (directed hypergraphs). Examples of the
former are elections, where a subset of voters is an elec-
tion an the collection of elections constitutes the hyper-
graph. Chemical reactions [3, 22, 23] and particle scatter-
ings are instances of directed hypergraphs, where some
starting materials are transformed into some products.
For hypergraphs, in principle, various generalizations of
the Forman-Ricci graph curvature are possible. It is a
main contribution of this paper to identify that notion
of Forman-Ricci curvature for (un)directed hypergraphs
that is best adapted to their structure and to investigate
its properties. We also apply this to concrete empirical
hypernetworks, a social and a metabolic one.
In this section we briefly summarize the results of the
Forman-Ricci curvature for graphs and then generalize
the curvature for hypergraphs.
A. Undirected graphs
Let G= (V, E ) be a (multi)graph with vertex set V
and multiset of edges E. The Forman-Ricci curvature of
an edge e={i, j} ∈ E, as introduced in [11], is given by:
F(e) = we wi
we
+wj
weX
eli
wi
wewelX
elj
wj
wewel!(1)
where wedenotes the weight of the edge e,wiand wj
are the weights of vertices iand j, respectively. The
sums over elkrun over all edges elincident on the
vertex kexcluding e. The curvature for the unweighted
multigraph, with vertex and edge weights set to 1, is
given by [2]
F(e)=4didj(2)
where dkis the vertex degree of k. Defining D=Pkedk
we have
F(e) = 4 D(3)
As a multigraph may have repeated edges, whose num-
ber is independent of the number of vertices, the bounds
for F(e) shall be expressed as a function of the known
number of edges, namely, |E|. Therefore, 2(2 − |E|)
F(e)2. The lower bound is attained when dk=|E|
for every ke, therefore D= 2|E|(Figure 1a). In turn
F(e) = 2, for an isolated edge e(Figure 1c). In contrast
to the multigraph case, for simple unweighted graphs, the
lower bound can be expressed as a function of the num-
ber of vertices: 2(3 − |V|)F(e), which is obtained for
dk=|V|1 for every ke, i.e., D= 2(|V| 1) (Figure
1b). As for multigraphs, F(e) reaches its maximum value
(F(e) = 2) for an isolated edge (Figure 1c).
As seen in Figure 1, Forman-Ricci curvature quantifies
the degree of spread of the vertices in e, from maximum
spread (corresponding to min F(e)) to minimum spread
(attained when max F(e)).
min F max F
|E|=9
D=10
F=-6
|E|=1
D=2
F=2
a b c
|E|=5
D=10
F=-6
|E|=5
D=7
F=-3
|E|=4
D=5
F=-1
|E|=9
D=12
F=-8
|E|=3
D=5
F=-1
|E|=7
D=8
F=-4
|E|=2
D=3
F=1
FIG. 1. Forman-Ricci curvatures F(e) calculated for the red
edge eof the depicted undirected graphs.
B. Directed graphs
Here we are interested in an unweighted directed multi-
graph G= (V, E ), where e= (i, j)Eis an arc (directed
edge), and i, j V. Equation 2 indicates that the curva-
ture of an edge depends on the degree of its vertices. As
in a simple directed graph the degree can be split into
3
in- and out-degree. The curvature of e= (i, j) is defined
in terms of in- and out-degrees as well [17]. There are
different possibilities for the realization of the curvature,
depending on the meaning one assigns to it. Here we
emphasize the directed spread or flow through e, i.e., fol-
lowing the direction of the arc. Therefore, we consider
the incoming arcs on i(in-degree of i, in(i)) and the out-
going arcs from j(out-degree of j, out(j)). When we
separate the curvature in (2) into the contribution 2 di
of iand 2 djof jand also note that the edge ecounts
for the degrees of iand j, but neither for the in-degree of
inor for the out-degree of j, then a curvature accounting
for the in-flow at i(F(e)) and another for the out-flow
at j(F(e)) is defined as
F(e) = 1 in(i)
F(e) = 1 out(j).(4)
Both are bounded below by 2 − |E|for in(i) = out(j) =
|E|1, and bounded above by 1 when in(i) = out(j)=0
(Figure 2a). For the simple directed graph the lower
bound for both, in- and out-flow, is 2 − |V|, for in(i) =
out(j) = |V|−1 (Figure 2b). The upper bound is reached,
in both cases, when in(i) = out(j) = 0 (Figure 2c). The
curvature accounting for the flow through e= (i, j) is
then given by
F(e) = F(e) + F(e)
= 2 in(i)out(j)(5)
where 2(2 |E|)F(e)2 for the multigraph case
and 2(2 − |V|)F(e)2 in the simple graph case.
Figure 2c shows the case where F(e) = 2. Some
further examples of calculations of curvatures F(e)
are shown in Figure 2.
If the flow-loss along eis to be considered, two addi-
tional curvatures are calculated that account for the flow
loss at i(F(e)) and at j(F(e)). Thus
F(e)=1out(i)
F(e)=1in(j)(6)
both bounded below by 1 − |E|, for out(i) = in(j) =
|E|, and bounded above by 0 for out(i) = in(j) = 1
(Figure 2d). For the simple directed graph we have 2
|V| ≤ F(e)0 and 2 − |V| ≤ F(e)0. Hence, the
curvature for the flow-loss along e= (i, j) is
F(e) = F(e) + F(e)
= 2 out(i)in(j)(7)
where 2(1 − |E|)F(e)0 (Figures 2a-e) holds in
the multigraph case and 2(2|V|)F(e)0 in the
simple graph case. Some further examples are shown in
Figure 2.
A curvature accounting for the total flow over eis then
computed as
F(e) = F(e) + F(e) (8)
In the following section we extend the Forman-Ricci
curvature to hypergraphs.
FIG. 2. Forman-Ricci curvatures F(e), F(e), and
F(e) calculated for the red arc eof the depicted directed
graphs.
III. FORMAN-RICCI CURVATURE OF
HYPER(EDGES/ARCS) IN HYPERGRAPHS
Given a set of vertices V, a graph is a collection of
subsets (edges) of V, all of which comprise only two ele-
ments. If we call the cardinality of each subset its size,
then a graph is a collection of subsets of size two. In a
hypergraph, the size of the subsets is no longer restricted,
and subsets of any size are allowed.
A. Undirected hypergraphs
An undirected hypergraph H= (V, E) consists of a set
Vof vertices and a multiset Eof subsets of V, called
hyperedges, such that eV, i.e. |e|≤|V|, for eE.
Some examples of hypergraphs are shown in Figure 3.
Separating the contributions of vertices iand jin
Equation 1, it can be rewritten as:
F(e) = we" wi
weX
eli
wi
wewel!+ wj
weX
elj
wj
wewel!#
(9)
4
furthermore,
F(e) = we"X
ke wk
weX
elk
wk
wewel!# (10)
Since Equation 10 no longer restricts eto size two, we
present it as the Forman-Ricci curvature of the hyper-
edge e. For the unweighted hypergraph, where all vertex
weights are equal to 1, this expression simplifies to
F(e) = X
ke 2dk!= 2|e| − X
ke
dk= 2|e| − D(11)
which is bounded below by |e|(2|E|) when dk=|E|for
every ke, and bounded above by 1 when D=|e|. In
other words, the minimum curvature occurs when every
vertex in ebelongs to each hyperedge (Figures 3a,b); the
maximum is attained for an isolated hyperedge (Figure
3c).
For the particular case of simple hypergraphs, we
therefore have the lower bound 2|e|(1 2|V|−2) when
dk=|P(V\ {k})|for every ke, and the upper bound
|V|, when E={V}. Note that in hypergraphs |e|≤|V|,
therefore the minimum value |e|may reach 1, unlike
graphs. In such a case, 2(1 2|V|−2)F(e)≤ |V|.
Some further examples of curvature for hypergraphs are
shown in Figure 3.
FIG. 3. Forman-Ricci curvatures F(e) calculated for the blue
hyperedge eof the depicted hypergraphs.
B. Directed hypergraphs
In a directed hypergraph, each hyperedge is composed
of two subsets of vertices: the tail and the head of the
hyperedge. Formally, we say that a directed hypergraph
His the couple (V, E ) with Va set of vertices and Ea
multiset of hyperarcs. A hyperarc is a pair e= (ei, ej),
where eiVand ejVare called its tail and its head,
respectively. Figure 4 depicts some examples of diercted
hypergraphs, where the sets eiand ejare highlighted.
Starting from the definitions of curvature for an arc in
the directed graph case (Equation 4), we introduce the
curvatures F(e) and F(e) for a hyperarc as
F(e) = |ei| − X
iei
in(i)
F(e) = |ej| − X
jej
out(j)(12)
with bounds |ei|(1 − |E|)F(e)≤ |ei|and |ej|(1
|E|)F(e)≤ |ej|. For the simple directed hyper-
graphs, we have |ei|(1 2|V|−1)F(e)≤ |ei|and
|ej|(1 2|V|−1)F(e)≤ |ej|. With F(e) and F(e)
at hand, we define the curvature for the flow through
e= (ei, ej) as:
F(e) = F(e) + F(e)
=|ei|+|ej| − X
iei
in(i)X
jej
out(j)(13)
with bounds (1 |E|)(|ei|+|ej|)F(e)≤ |ei|+|ej|
in the general case and (12|V|)(|ei|+|ej|)F(e)
|ei|+|ej|for the simple directed hypergraph (Figure 4).
Note that if |e|is allowed to have its minimum value of
1, then |ek|= 1 and 2(1 − |E|)F(e)2. Some
examples of curvature values for directed hypergraphs are
shown in Figure 4.
The respective flow-loss curvatures are:
F(e) = |ei| − X
iei
out(i)
F(e) = |ej| − X
jej
in(j)(14)
with bounds |ei|(1 − |E|)F(e)0 and |ej|(1
|E|)F(e)0 in the general case and |ei|(1 2|V|)
F(e)0 and |ej|(1 2|V|)F(e)0 for the simple
directed hypergraphs.
Equation 14 yields the flow-loss curvature
F(e) = F(e) + F(e)
=|ei|+|ej| − X
iei
out(i)X
jej
in(j)(15)
with bounds (1 − |E|)(|ei|+|ej|)F(e)0, which
becomes (12|V|−1)(|ei|+|ej|)F(e)0 for simple
directed hypergraphs.
5
|E|=3
F(e)=-2
F(e)=-1
F(e)=0 (max)
F(e)=0 (max)
F(e)=-3
F(e)=0
F(e)=0
a
eiej
eiej
eiej
eiej
eiej
eiej
|E|=1
F(e)=0 (min)
F(e)=0 (min)
F(e)=0 (min, max)
F(e)=0 (min, max)
F(e)=0 (min)
F(e)=0 (min, max)
F(e)=0
b
|E|=1
F(e)=2 (max)
F(e)=1 (max)
F(e)=0 (min, max)
F(e)=0 (min, max)
F(e)=3 (max)
F(e)=0 (min)
F(e)=3
c
|E|=3
F(e)=2 (max)
F(e)=1 (max)
F(e)=-1
F(e)=-2 (min)
F(e)=3 (max)
F(e)=-3
F(e)=0
d
|E|=6
F(e)=-2
F(e)=-1
F(e)=-1
F(e)=-1
F(e)=-3
F(e)=-2
F(e)=-5
e
|E|=5
F(e)=0
F(e)=0
F(e)=-1
F(e)=0 (max)
F(e)=0
F(e)=-1
F(e)=-1
f
|E|=5
F(e)=1
F(e)=0
F(e)=-2
F(e)=0 (max)
F(e)=1
F(e)=-2
F(e)=-1
g
ei=ej
FIG. 4. Forman-Ricci curvatures F(e), F(e), and
F(e) calculated for the red hyperarc e, connecting vertices
in eiwith those in ej, of the depicted hypergraphs.
In the following section we calculate the Forman-Ricci
curvature for different cases that can be modelled as hy-
pergraphs. Several applications of the Forman-Ricci cur-
vature for the graph case are found in references [2, 11–
19].
APPLICATIONS TO EMPIRICAL NETWORKS
C. Wikipedia Voting Network
Wikipedia is an encyclopedia written by volunteers. A
small part of these users are administrators, who besides
being active, regular long-term Wikipedia contributors,
have gained the general trust of the community and have
taken on technical maintenance duties. A user becomes
an administrator when a request for adminship is issued
and the Wikipedia community via a public vote decides
who to promote to administrator. Users can either sub-
mit their own requests for adminship or may be nomi-
nated by other users. Using the January 3 2008 dump
of Wikipedia page edit history [24], Leskovec et al. [25]
extracted 2,794 elections (hyperedges in our setting) and
7,066 users (vertices) participating in the elections (ei-
ther casting a vote or being voted on). We calculated
the curvature for the resulting undirected hypergraph.
Figure 5 shows the distribution of hyperedge size and of
vertex degree. The data show that many of the elections
involve a single user, although elections with 2-20 users
are also common. There are few elections with more than
100 users, the largest one including 370 users (Figure 5a).
The participation in elections is heavy-tailed distributed
(Figure 5b), with most of the users participating in a
single election and very few taking part in about a thou-
sand elections. The curvature values are mostly negative
(Figure 5c), indicating (i) the absence of elections with
unexperienced users (max F(e)6=|e|), i.e., all elections
at least include a user that takes part in at least one
other election; and (ii) for most elections the number of
elections in which users take part is greater than their
number of voting users (D > |e|in Equation 3). The
minimum curvature value (-3,112) is far from the lower
bound (19,728,272, calculated with |e|= 7,066). This
reflects the fact that most users are experts in limited
fields only.
To have some insight about the effects of hyperedge
sizes and number of incident hyperedges on curvature, we
analyzed their distributions over the span of curvature
values (Figures 5d-f). Figure 5d shows that the more
spread the election, i.e. involving users that vote in other
elections, the larger the number of users voting. Figures
5e and f show that, in average, elections overlap with a
low number of other elections (low number of incident
elections). Thus, the curvature values are mainly ruled
by hyperedge size rather than by incident hyperedges.
D. Metabolic Network of Escherichia coli
The metabolism of Escherichia coli is one of the most
studied and best characterized among bacteria. Here we
model the metabolism K-12 (iJR904 GSM/GPR) [26] of
this bacterium as a directed hypergraph whose vertices
are the metabolites (chemical species). Each chemical
reaction is represented as a hyperarc e, whose educts
(starting materials) correspond to eiand products to ej.
There are |V|= 625 metabolites and |E|= 1,176 reac-
tions accounting for 686 non-reversible and 245 reversible
ones. These latter reactions, denoted by eiejhave
been included as “forward” (eiej) and “backward”
(ejei) reactions. All curvatures (Equations 12 to 15)
and related calculations are gathered in the Supplemen-
tary Material.
As expected for chemical reactions, typically there are
not more than three educts and three products (Figure
6a). The curvature values therefore vary little in response
to hyperarc size, but rather depend more on the degree
of vertices in eiand ej. Note that these degrees result,
6
a)
100101102
100
101
102
Hyperedge degree
Frequency
b)
100101102103
100
101
102
103
Vertex degree
Frequency
c)
-30000 -25000 -20000 -15000 -10000 -5000 0
100
101
102
Frequency
d)
-30000 -25000 -20000 -15000 -10000 -5000 0
0
0.2
0.4
0.6
0.8
1
Normalized hyperedge size
e)
-30000 -25000 -20000 -15000 -10000 -5000 0
0
0.2
0.4
0.6
0.8
1
Curvature
Median degree
f)
-30000 -25000 -20000 -15000 -10000 -5000 0
0
0.2
0.4
0.6
0.8
1
Curvature
Average degree
FIG. 5. Voting Wikipedia: Distribution of a) hyperedge size
(size of elections) and b) vertex degree (participation of users
in elections). c) Histogram of curvature values with bins of
10 units. Box-plots of d) normalized hyperedge sizes, e) me-
dian, and f) average hyperedge degrees corresponding to each
curvature bin of c).
respectively, from the summation over vertex degrees of
educts and of products (Equations 12 to 15). The dis-
tribution and educts and products degrees is shown in
Figure 6. The participation of educts and products in
reactions does not yield a smooth distribution, as indi-
cated by the gaps present in Figure 6b,c. The production
of educts (Figure 6b) shows a large group of reactions
whose educts are synthesized by less than 200 reactions
and another group where they are obtained by more than
450 reactions. Likewise, there are two groups of reactions
with different levels of use of their products (Figure 6c);
one group has reactions whose products are used in less
than 100 reactions and another with more than 300 re-
actions taking their products as starting materials.
The synthesis of products and the use of educts (Fig-
ures 6d and e), shows also a discontinuous participation
of substrates in reactions. There are two groups of re-
actions according to the number of reactions synthesiz-
ing their products: one with reactions whose products
are obtained by less than 200 reactions and another by
more than 450 reactions (Figure 6d). Likewise, there are
various groups of reactions according to the use of their
educts, from some which are seldom used to some others
with about 170, 230, and more than 330 uses (Figure
6e).
a)
0.5 1 1.522.533.544.5 5 5.5 6
1
2
3
4
5
6
7
8
7
63
103
6
9
34
169
492
1
1
66
95
98
1
4
4
7
11
1
1
3
|ei|
|ej|
b)
0 100 200 300 400 500 600 700 800
100
101
102
c)
0 100 200 300 400 500 600 700 800
100
101
102
d)
0 100 200 300 400 500 600 700 800
100
101
102e)
0 100 200 300 400 500 600 700 800
100
101
102
FIG. 6. Metabolic network: a) Scatter plot of sizes of educts
(|ei|) and products (|ej|), where circle radii correspond to
log f / log 100, being fthe frequency of appearance of the cou-
ple (|ei|,|ej|) in the reactions. Numbers inside circles corre-
spond to f. Distribution of b) Pieiin(i), c) Pieiout(i),
d) Pjejin(j), and Pjejout(j).
The extent to which the educts of the reaction eare
produced from other reactions is measured by F(e).
The more reactions lead to the educts of e, the more
negative F(e) becomes (Figure 7a). The theoreti-
cal bounds of F(e), assuming max |ei|= 625 are
734,375 F(e)625. However, more realistic
bounds are 7,050 F(e)6, which results from
taking the actual max |ei|= 6 (Figure 6a). We found
that min F(e) = 735, which is attained by four re-
actions, with four educts (all substrate abbreviations are
included in the Appendix (Table I)):
adp+h+malcoa+piaccoa+atp+hco3
adp+h+pi+25aicsasp L+atp+5aizc
adp+dtbt+h+piatp+co2+dann
atp+gar+h+piatp+gly+pram
These reactions are those whose educts are the most
synthesized of all the metabolic reactions of E. coli (63%
of the reactions produce their educts). In three of them
atp is synthesized from adp, which shows the well-known
7
central metabolic role of atp [27, 28].
max F(e) = 1 corresponds to a single reac-
tion: cyan+tsulh+so3+tcynt, where only one of
its two educts is a product of a single reaction:
atp+h2o+tsuladp+h+pi+tsul.
Figure 7a shows that the most frequent curvature value
is 0 (for 73 reactions), i.e. 6% of the reactions have a
trade-off between the number of educts and the num-
ber of reactions producing them; most of the remaining
reactions have more ways to produce their educts than
the number of educts. It is also found that there are
almost no reactions with curvatures between -200 and
-450, indicating that educts of reactions are mainly ob-
tained either by less than 200 reactions (less than 17% of
the reactions) or by 450 to 600 reactions (38 to 51% of
the reactions). This is a consequence of the heavy-tailed
in-degree distribution of substrates [27].
Figure 7b shows the curvature values F(e), which
quantify the extent to which products of reactions are
used in further reactions as educts. By taking max |ej|=
8 (Figure 6a) this curvature takes values 9,400
F(e)8. The actual min F(e) = 729, for
adp+h+piatp+h+h2o, i.e., this is the reaction whose
three products are most used in other reactions as start-
ing materials (used in 62% of the reactions). In contrast,
there are four reactions with max F(e) = 1:
agpe EC +pg EC apg EC+g3pe
agpc EC +pg EC apg EC+g3pc
agpg EC +pg EC apg EC+g3pg
udpgal udpgalfur
Hence, for those three reactions with two products, these
substrates are only used in a further reaction as educts,
while udpgalfur is not further used, i.e. it is a metabolic
“dead-end” [26]. As most of the reactions (96%) have
negative values of F(e), this indicates the efficient use
of reaction products [28], which can be divided into two
regimes. For about half of the reactions their products
are used in no more than 9% of the reactions and about
40% of the reactions have products that are used in more
than a quarter of the reactions. This is a consequence
of the heavy-tailed distribution, this time, of the out-
degrees of the substrates [27].
F(e) showed that for most of the reactions their
educts are produced by other reactions and F(e) that
the products are used in several other reactions. The
question that arises whether those popular educts are
connected through reactions with the popular products
is positively answered by F(e), which takes negative
values for most of the reactions. The min F(e) =
1,463 corresponds to adp+h+piatp+h+h2o. Hence,
this is the reaction whose educts are most synthesized by
other reactions and whose products are the most used
as educts in other reactions. It is the bottleneck of the
E. coli metabolism. Other reactions of this sort, with
F(e)<1,000 (Figure 7e), are:
adp+h+piatp+h+h2o
h+o2+q8h2h+h2o+q8
h+o2+q8h2h+h2o+q8
h+no3+q8h2h+h2o+no2+q8
h+mql8+no3h+h2o+mqn8+no2
Having analyzed the metabolism following the direc-
tion of educts to products in reactions, we now proceed
to study the curvature in the backward direction, which
quantifies to which extent a reaction is just one of the
many connecting popular educts with popular products.
We start by analyzing F(e) that shows to which ex-
tent educts of a reaction participate in other reactions.
The theoretical bounds are 734,375 F(e)0 and
we found that F(e) takes values in between -729 and
0; the minimum is attained by atp+h+h2oadp+h+pi,
indicating that atp in an acidic aqueous medium is the
most often used starting material. max F(e) occurs for
51 reactions, whose involved 56 educts are only used in
those 51 reactions, i.e. they are very specialized educts
for very particular metabolic reactions. The distribution
of F(e) values shows that for half of the reactions, their
educts participate in less than 9% of the reactions, while
for the rest, their educts take part in more than 15% of
the reactions.
F(e) shows to which extent products of a reaction
are synthesized by other reactions. The theoreti-
cal bounds are given by max |ej|= 8, leading to
9,400 F(e)0. The actual values range from
-788 to 0. The minimum is reached by reaction:
dxyl5p+nad+phthrco2+h+h2o+nadh+pdx5p+pi,
i.e. this set of products is the most synthesized by E. coli
metabolism, which is expected, for the likelihood of a
set of substances to be synthesized scales with the size of
the set. This reaction with six products is one of the few
where more than the frequent one to four products are
synthesized (Figure 6a). Moreover, among the products,
co2,h,h2o,nadh, and pi are often products of other
reactions.
max F(e) = 0 is attained by 29 reactions, all of them
leading to a single product, except for three reactions,
each one with two products. Thus, those 32 products are
of little synthetic relevance for the metabolism. The dis-
tribution of curvature values shows that there are three
kinds of reactions whose products are synthesized by dif-
ferent number of reactions. For 60% of the reactions their
products are synthesized by less than 200 reactions (17%
of the reactions) and for the rest of the reactions by more
than 450 reactions (38% of the reactions).
Curvatures F(e) and F(e) showed that half of the
educts are often used and 40% of the products are of-
ten synthesized, which indicates that it is very likely to
find alternative ways to link educts with products of ex-
isting reactions, as found in [27–29]. A measure of this
degree of redundancy of a reaction or of its replaceability
is given by F(e), which indicates to which extent a
reaction connects popular educts with popular products.
The more negative the curvature, the more redundant or
likely replaceable the reaction is.
By analyzing F(e) distribution (Figure 7f) it is
seen an ample spectrum of curvatures, with almost no
gaps, indicating different degrees of redundancy for the
8
metabolic reactions. min F(e) = 1,463 corresponds
to the hydrolysis of ATP, i.e., atp+h+h2oadp+h+pi,
indicating, e.g., that the dephosphorilation of atp to adp
can be achieved by many other reactions (12% of the re-
actions). max F(e) = 0 occurs for the following eight
reactions, which are unique as they are the only way to
connect their educts with their products:
mmcoa Rmmcoa S
5mdr1p5mdru1p
gdpddmangdpofuc
adphep D,Dadphep L,D
dhnptgcald+6hmhpt
glu1sa5aop
prfpprlp
pran2cpr5p
a)
800 700 600 500 400 300 200 100 0
100
101
b)
800 700 600 500 400 300 200 100 0
100
101
c)
800 700 600 500 400 300 200 100 0
100
101
d)
800 700 600 500 400 300 200 100 0
100
101
e)
1,6001,4001,2001,000 800 600 400 200 0
100
101
f)
1,6001,4001,2001,000 800 600 400 200 0
100
101
1
FIG. 7. Metabolic network: Histograms of curvature values
for a) F(e), b) F(e), c) F(e), d) F(e), e) F(e),
and f) F(e) with bins of 500 units.
CONCLUSIONS AND OUTLOOK
The Forman-Ricci curvature emphasizes the impor-
tance of the relational character of (hyper)edges, thereby
providing a view of the network structure that comple-
ments traditional vertex-centered descriptors. It also em-
beds the characterization in a formal mathematical the-
ory, namely Riemannian geometry.
The results here reported include a brief review of
Forman-Ricci curvature for (un)directed graphs and gen-
eralize the curvature to both undirected and directed hy-
pergraphs. Graph curvatures used in previous studies
thus become particular cases of the curvature for hyper-
graphs [2, 11–19]. We determined the upper and lower
bounds for Forman-Ricci curvature for graphs and hy-
pergraphs, which so far had not been studied.
The curvatures here presented aim at quantifying the
trade-off between hyperedge(arc) size and the degree of
participation of vertices members of the hyperedge(arc)
in other hyperedges(arcs). For undirected hypergraphs,
the curvature takes negative values when the degree of
vertices of the hyperedge is more significant than the size
of the hyperedge. For directed hypergraphs we devised
four curvatures that gauge different aspects of hyperarcs.
F(e) quantifies the trade-off between the size of the hy-
perarc tail and the input of its vertices, F(e) do so for
the size of the hyperarc head and the output of its ver-
tices; while F(e) and F(e) consider the size of the tail
and the output of their vertices, and the size of the head
and the input of their vertices, respectively. These cur-
vatures are combined into F(e) and F(e), which
account for the flow through hyperarc eand for its re-
dundancy, respectively.
The Forman-Ricci curvature for hypergraphs intro-
duced here differs from the alternative construction pro-
posed in [30]. There, hyperedges are interpreted as sim-
plices. Here, we focus entirely on hyperedges, their
sizes, and the degrees of vertices, thereby avoiding the
re-interpretation of hyperedges as “higher-dimensional”
objects and implicitly introducing additional structures,
like boundaries of simplices, that are not part of the orig-
inal data. For the particular case of directed hypergraphs
we disentangled the curvature in the four aforementioned
informative measures that allow a detailed exploration
of the hypergraph structure. Moreover, we applied these
curvatures to the analysis of two large networks, one of
social and the other of chemical interactions.
The analysis of Wikipedia vote network exemplified
the Forman-Ricci curvature of undirected hypergraphs,
where elections constituted hyperedges and users/voters
vertices. We found that curvature is mostly ruled by
hyperedge size rather than by hyperedge degree. Like-
wise, the more users involved in elections, the more the
presence of experienced users. In a traditional graph set-
ting [24, 25], with users as vertices and votes as couple
of users, conclusions such as the previous one on elec-
tions cannot be drawn. This shows the richness of hy-
pergraphs and their curvatures, which for this particular
case allowed the definition of a hyperedge as an election.
Forman-Ricci curvature for directed hypergraphs was
computed over the metabolic network of E. coli, which
traditionally has been analyzed through a graph setting
[27–29] and which has shed light on the important role
of several substrates for the metabolic stability. In our
approach, rather than focusing on substrates, we did on
reactions, which were characterized as hyperarcs connect-
ing sets of educts with sets of products.
9
In contrast to the Wikipedia vote network, we found
that curvature values for the metabolic network were
ruled by the degree of hyperarcs, i.e. of in- and out-
degrees of tails and heads of hyperarcs. This is a chem-
ical consequence, for it is unlikely that several educts
collide simultaneously to give place to a product. In fact
reactions where more than five educts participate in a
single-step reaction are scarce [31].
We emphasize that the strong dependence of hyperarc
curvature is on the summation of the degrees of the vertex
belonging to the hyperarc, which is different from the
traditional degree of isolated vertices.
With curvature results at hand we defined “bottle
neck” reactions as those few reactions whose educts are
readily available (obtained from several reactions) and
whose products are often used as educts. They are char-
acterized by having very negative F(e) values. For
E. coli this reactions is: adp+h+piatp+h+h2o. Bot-
tle neck reactions can be considered as assortative ones,
for they transform popular products into popular educts.
Curvature values also allowed detecting redundant re-
actions (“one of the crow reactions”), which can be easily
replaced by others. The suitable curvature for detecting
such reactions is F(e), whose most negative values
correspond to reactions where popular sets of educts are
connected to popular sets of products. For E. coli, this
reaction is atp+h+h2oadp+h+pi. Thus, adp phos-
phorilation is the metabolic bottle neck reaction but the
reverse reaction is not that central for the metabolism.
Our results show that E. coli metabolic network makes
use of a wealth amount of the products of its reactions
to start other reactions. This contrast with the historical
trend in wet-lab chemistry reactions, where most of the
products are seldom used in further reactions [32]. As the
historical study was conducted over single substances,
rather than over educts and products, further work on
the curvature of wet-lab chemical reactions needs to be
done to determine whether the behaviour found for E.
coli is also a trend of chemical reactions, in general.
The curvatures here presented, as indicated in Equa-
tion 1 and as used in [2, 11–19], can be weighted. In
the recent sketch of curvature for hypergraphs [30], the
weights are calculated based on the volume of the sim-
plex associated to the hyperedge. Weights, however, can
also be based on meta information of the network, e.g.
user’s seniority in the Wikipedia example or stoichiomet-
ric coefficients in the metabolic network. This and other
weighting schemes need to be explored in future stud-
ies on the curvature of hypergraphs, which our approach
allows.
ACKNOWLEDGMENTS
WL was supported by a PhD scholarship from
the German Academic Exchange Service (DAAD):
Forschungsstipendien-Promotionen in Deutschland,
2017/2018 (Bewerbung 57299294).
IV. APPENDIX
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11
TABLE I. Partial list of substrates of the E. coli metabolic
network.
Abbreviation Chemical name
25aics (S)-2-[5-Amino-1-(5-phospho-D-
ribosyl)imidazole-4-carboxamido]succinate
5aizc 5-amino-1-(5-phospho-D-ribosyl)imidazole-4-
carboxylate
5aop 5-Amino-4-oxopentanoate
2cpr5p 1-(2-Carboxyphenylamino)-1-deoxy-D-
ribulose
5-phosphate
6hmhpt 6-hydroxymethyl dihydropterin
5mdr1p 5-Methylthio-5-deoxy-D-ribose 1-phosphate
5mdru1p 5-Methylthio-5-deoxy-D-ribulose 1-phosphate
accoa Acetyl-CoA
adp Adenosine diphosphate
adphep D,D ADP-D-glycero-D-manno-heptose
adphep L,D ADP-L-glycero-D-manno-heptose
asp L L-Aspartate
atp Adenosine triphosphate
co2 Carbon dioxyde
cyan Hydrogen cyanide
dann 7,8-Diaminononanoate
dhnpt Dihydroneopterin
dtbt Dethiobiotin
dxyl5p 1-deoxy-D-xylulose 5-phosphate
tsul Thiosulfate
gar N1-(5-Phospho-D-ribosyl)glycinamide
gcald Glycolaldehyde
gdpddman GDP-4-dehydro-6-deoxy-D-mannose
gdpofuc GDP-4-oxo-L-fucose
glu1sa L-Glutamate 1-semialdehyde
gly Glycine
hH+
h2o Water
hco3 Bicarbonate
malcoa Malonyl CoA C24H33N7O19P3S
mmcoa R (R)-Methylmalonyl-CoA
mmcoa S (S)-Methylmalonyl-CoA
mql8 Menaquinol 8
mqn8 Menaquinone 8
nad Nicotinamide adenine dinucleotide
nadh Nicotinamide adenine dinucleotide - reduced
no2 Nitrite
no3 Nitrate
o2 Molecular oxygen
pdx5p Pyridoxine 5’-phosphate
phthr O-Phospho-4-hydroxy-L-threonine
pi Phosphate
pram 5-Phospho-beta-D-ribosylamine
pran N-(5-Phospho-D-ribosyl)anthranilate
prfp 1-(5-Phosphoribosyl)-5-[(5-
phosphoribosylamino)methylideneamino]imidazole-
4-carboxamide
prlp 5-[(5-phospho-1-deoxyribulos-1-
ylamino)methylideneamino]-1-(5-
phosphoribosyl)imidazole-4-carboxamide
q8 Ubiquinone-8
q8h2 Ubiquinol-8
so3 Sulfite
tcynt Thiocyanate
tsul Thiosulfate
1
SUPPLEMENTARY MATERIAL
Curvature values for chemical reactions from
the metabolism of Escherichia coli K-12 (iJR904
GSM/GPR).
Reaction F(e)F(e)F(e)F(e)F(e)F(e)
ALATA_L-24 -42 -66 -30 -77 -107
ALATA_L-77 -30 -107 -42 -24 -66
ALAR-7 -5 -12 -10 -4 -14
ALAR-4 -10 -14 -5 -7 -12
ASNN -90 -24 -114 -236 -41 -277
ASNS2 -78 -340 -418 -196 -558 -754
ASNS1 -131 -368 -499 -434 -592 -1026
ASPT -7 -17 -24 -17 -44 -61
ASPTA-24 -32 -56 -37 -40 -77
ASPTA-40 -37 -77 -32 -24 -56
VPAMT -8 -17 -25 -13 -47 -60
DAAD -91 -21 -112 -245 -84 -329
ALARi -7 -5 -12 -10 -4 -14
FFSD -86 -6 -92 -233 -8 -241
A5PISO-3 -1 -4 -3 0 -3
A5PISO0 -3 -3 -1 -3 -4
MME-1 -1 -2 0 0 0
MME0 0 0 -1 -1 -2
MICITD -86 0 -86 -233 -1 -234
ALCD19-498 -55 -553 -364 -45 -409
ALCD19-45 -364 -409 -55 -498 -553
LCADi -130 -366 -496 -287 -499 -786
TGBPA-1 -16 -17 0 -21 -21
TGBPA-21 0 -21 -16 -1 -17
LCAD-130 -366 -496 -287 -499 -786
LCAD-499 -287 -786 -366 -130 -496
ALDD2x -132 -365 -497 -287 -508 -795
ARAI-3 -1 -4 -3 0 -3
ARAI0 -3 -3 -1 -3 -4
RBK_L1 -37 -358 -395 -173 -590 -763
RBP4E-2 -3 -5 0 -4 -4
RBP4E-4 0 -4 -3 -2 -5
ACACCT -14 -3 -17 -18 -14 -32
BUTCT -14 -3 -17 -18 -12 -30
AB6PGH -85 -3 -88 -233 -8 -241
PMANM0 -2 -2 -1 -3 -4
PMANM-3 -1 -4 -2 0 -2
PPM2-5 -1 -6 -5 0 -5
PPM20 -5 -5 -1 -5 -6
PPM-5 -3 -8 -5 -8 -13
PPM-8 -5 -13 -3 -5 -8
DRPA 0 -13 -13 -1 -17 -18
GALCTND -1 -233 -234 -2 -86 -88
DDPGALA-1 -23 -24 0 -55 -55
DDPGALA-55 0 -55 -23 -1 -24
DDGALK -37 -358 -395 -172 -589 -761
DHAPT -6 -21 -27 -23 -52 -75
FAO4 -128 -362 -490 -290 -505 -795
ALDD19x -127 -362 -489 -283 -496 -779
FRUK -37 -359 -396 -172 -590 -762
FCLPA-1 -11 -12 0 -12 -12
FCLPA-12 0 -12 -11 -1 -12
FCI-2 -1 -3 -2 0 -2
FCI0 -2 -2 -1 -2 -3
FCLK -37 -358 -395 -173 -589 -762
LCAR-499 -52 -551 -366 -43 -409
LCAR-43 -366 -409 -52 -499 -551
UDPG4E-3 -2 -5 -5 -1 -6
UDPG4E-1 -5 -6 -2 -3 -5
GALKr-41 -359 -400 -174 -589 -763
GALKr-589 -174 -763 -359 -41 -400
UGLT-4 -11 -15 -6 -7 -13
UGLT-7 -6 -13 -11 -4 -15
GALU-456 -12 -468 -339 -75 -414
GALU-75 -339 -414 -12 -456 -468
GALCTD -1 -233 -234 -2 -87 -89
GLTPD-42 -363 -405 -50 -496 -546
GLTPD-496 -50 -546 -363 -42 -405
GLYCTO2 -8 -6 -14 -17 -18 -35
GLYCTO3 -11 -8 -19 -13 -13 -26
GLYCTO4 -10 -7 -17 -9 -8 -17
GLYCDx -45 -365 -410 -55 -499 -554
GLYK -41 -365 -406 -177 -597 -774
PGLYCP -85 -40 -125 -233 -151 -384
G3PD2-53 -374 -427 -25 -475 -500
G3PD2-475 -25 -500 -374 -53 -427
GLCRAL -1 -15 -16 0 -45 -45
DHPPD -41 -362 -403 -50 -497 -547
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arXiv:1811.07825v1 [cs.DM] 19 Nov 2018
2
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HISTP -86 -36 -122 -233 -146 -379
HSTPT -34 -20 -54 -28 -17 -45
HISTD -127 -364 -491 -283 -499 -782
IG3PS -1 -353 -354 -12 -485 -497
ATPPRT -38 -7 -45 -183 -72 -255
PRATPP -86 -331 -417 -233 -521 -754
PRAMPC -86 0 -86 -233 0 -233
PRPPS-45 -341 -386 -175 -483 -658
PRPPS-483 -175 -658 -341 -45 -386
ACCOACr-51 -395 -446 -192 -735 -927
ACCOACr-735 -192 -927 -395 -51 -446
ACACT1r-13 -21 -34 -16 -25 -41
ACACT1r-25 -16 -41 -21 -13 -34
CDAPPA_EC -88 -330 -418 -236 -463 -699
DASYN_EC-454 -10 -464 -331 -74 -405
DASYN_EC-74 -331 -405 -10 -454 -464
CLPNS_EC-1 -5 -6 -4 -4 -8
CLPNS_EC-4 -4 -8 -5 -1 -6
C140SN -467 -268 -735 -382 -206 -588
C120SN -467 -267 -734 -382 -205 -587
MACPD -449 -9 -458 -334 -59 -393
KAS14 -450 -20 -470 -335 -76 -411
MCOATA-18 -31 -49 -8 -23 -31
MCOATA-23 -8 -31 -31 -18 -49
C160SN -467 -266 -733 -382 -206 -588
KAS16 -466 -34 -500 -378 -120 -498
C181SN -467 -266 -733 -382 -206 -588
C141SN -467 -266 -733 -382 -206 -588
KAS15 -462 -34 -496 -350 -82 -432
ACOATA-30 -22 -52 -23 -24 -47
ACOATA-24 -23 -47 -22 -30 -52
C161SN -467 -267 -734 -382 -206 -588
FAO2 -194 -391 -585 -485 -621 -1106
FAO1 -194 -391 -585 -485 -621 -1106
FAO3 -188 -391 -579 -484 -621 -1105
PGPP_EC -86 -40 -126 -234 -147 -381
PGSA_EC-11 -330 -341 -10 -460 -470
PGSA_EC-460 -10 -470 -330 -11 -341
PASYN_EC -14 -8 -22 -10 -20 -30
PSD_EC -449 -9 -458 -325 -59 -384
PSSA_EC-9 -330 -339 -12 -460 -472
PSSA_EC-460 -12 -472 -330 -9 -339
AHC-92 -7 -99 -234 -7 -241
AHC-7 -234 -241 -7 -92 -99
DHPTDC 0 -232 -232 0 -86 -86
RHCCE 0 -1 -1 0 -2 -2
HSST -2 -21 -23 -6 -23 -29
SHSL1 -2 -336 -338 -6 -470 -476
CYSTL -86 -22 -108 -233 -79 -312
METS -2 -5 -7 -1 -7 -8
METAT -125 -50 -175 -406 -220 -626
AHCYSNS -92 -5 -97 -234 -6 -240
LGTHL -3 0 -3 -2 0 -2
GLYOX -86 -329 -415 -233 -455 -688
MGSA -9 -36 -45 -7 -146 -153
UGLYCH -535 -18 -553 -557 -96 -653
ALLTN -87 -324 -411 -235 -449 -684
ALLTAH -86 -1 -87 -233 -3 -236
CYNTAH -450 -15 -465 -329 -92 -421
CMPN -97 -4 -101 -238 -9 -247
ADA -540 -11 -551 -563 -39 -602
DADA -537 -8 -545 -559 -36 -595
ADNK1 -42 -364 -406 -178 -621 -799
ADK1-70 -34 -104 -178 -139 -317
ADK1-139 -178 -317 -34 -70 -104
DADK-38 -35 -73 -173 -141 -314
DADK-141 -173 -314 -35 -38 -73
ADK4-33 -34 -67 -6 -139 -145
ADK4-139 -6 -145 -34 -33 -67
ADK3-34 -38 -72 -17 -146 -163
ADK3-146 -17 -163 -38 -34 -72
AMPN -119 -8 -127 -239 -14 -253
AP4AH -85 -358 -443 -233 -588 -821
GP4GH -85 -328 -413 -233 -456 -689
6
AP5AH -85 -530 -615 -233 -625 -858
ADPT -7 -13 -20 -16 -105 -121
CYTD -538 -11 -549 -561 -39 -600
DCYTD -536 -9 -545 -558 -37 -595
CYTK2-38 -35 -73 -173 -141 -314
CYTK2-141 -173 -314 -35 -38 -73
CYTK1-48 -36 -84 -177 -141 -318
CYTK1-141 -177 -318 -36 -48 -84
UMPK-44 -36 -80 -173 -145 -318
UMPK-145 -173 -318 -36 -44 -80
CSND -536 -12 -548 -558 -39 -597
ADNCYC -37 -6 -43 -172 -72 -244
DCTPD -536 -8 -544 -559 -36 -595
TMDPP-150 -5 -155 -40 -5 -45
TMDPP-5 -40 -45 -5 -150 -155
DURIPP-149 -10 -159 -38 -10 -48
DURIPP-10 -38 -48 -10 -149 -159
NTPTP1 -87 0 -87 -235 -4 -239
NTPTP2 -87 -1 -88 -244 -4 -248
DUTPDP -88 -333 -421 -234 -523 -757
GK1-42 -38 -80 -174 -146 -320
GK1-146 -174 -320 -38 -42 -80
DGK1-38 -35 -73 -173 -141 -314
DGK1-141 -173 -314 -35 -38 -73
XPPT -5 -8 -13 -15 -73 -88
HXPRT -5 -10 -15 -15 -75 -90
GUAPRT -5 -9 -14 -17 -77 -94
INSK -42 -361 -403 -176 -591 -767
GSNK -40 -360 -400 -174 -593 -767
NTPP3 -87 -332 -419 -235 -522 -757
NTPP4 -88 -336 -424 -239 -532 -771
NTPP5 -87 -332 -419 -234 -522 -756
NTPP6 -123 -337 -460 -405 -554 -959
NTPP7 -86 -332 -418 -235 -524 -759
NTPP8 -87 -332 -419 -239 -528 -767
NTPP1 -87 -332 -419 -235 -522 -757
NTPP2 -87 -333 -420 -244 -526 -770
NDPK1-44 -45 -89 -176 -140 -316
NDPK1-140 -176 -316 -45 -44 -89
NDPK2-43 -40 -83 -174 -140 -314
NDPK2-140 -174 -314 -40 -43 -83
NDPK3-39 -40 -79 -174 -141 -315
NDPK3-141 -174 -315 -40 -39 -79
NDPK5-39 -36 -75 -173 -140 -313
NDPK5-140 -173 -313 -36 -39 -75
NDPK6-39 -35 -74 -173 -141 -314
NDPK6-141 -173 -314 -35 -39 -74
NDPK7-39 -36 -75 -173 -140 -313
NDPK7-140 -173 -313 -36 -39 -75
NDPK8-39 -35 -74 -173 -140 -313
NDPK8-140 -173 -313 -35 -39 -74
NDPK4-39 -36 -75 -173 -139 -312
NDPK4-139 -173 -312 -36 -39 -75
RNDR1 -139 -234 -373 -42 -96 -138
RNDR2 -7 -234 -241 -12 -96 -108
RNDR4 -6 -234 -240 -10 -96 -106
RNDR3 -2 -234 -236 -10 -96 -106
RNTR1 -37 -234 -271 -180 -95 -275
RNTR2 -1 -235 -236 -19 -95 -114
RNTR3 -2 -235 -237 -14 -95 -109
RNTR4 -1 -234 -235 -14 -96 -110
URIDK2r-39 -35 -74 -174 -141 -315
URIDK2r-141 -174 -315 -35 -39 -74
DURIK1 -40 -360 -400 -174 -590 -764
TMDK1 -41 -359 -400 -176 -591 -767
TMDS -4 -1 -5 -5 -5 -10
DTMPK-40 -35 -75 -173 -141 -314
DTMPK-141 -173 -314 -35 -40 -75
URIK2 -6 -329 -335 -15 -463 -478
CYTDK2 -4 -333 -337 -15 -467 -482
PYNP2r-151 -10 -161 -40 -10 -50
PYNP2r-10 -40 -50 -10 -151 -161
UPPRT -6 -8 -14 -16 -79 -95
NTD1 -88 -38 -126 -235 -149 -384
NTD5 -89 -40 -129 -234 -150 -384
NTD6 -87 -38 -125 -234 -148 -382
NTD8 -87 -37 -124 -234 -149 -383
NTD3 -87 -37 -124 -234 -147 -381
NTD4 -97 -40 -137 -238 -149 -387
NTD7 -119 -42 -161 -239 -151 -390
NTD9 -91 -38 -129 -235 -149 -384
NTD11 -89 -40 -129 -236 -151 -387
NTD10 -87 -38 -125 -234 -149 -383
NTD2 -93 -40 -133 -234 -151 -385
PUNP6-148 -9 -157 -37 -9 -46
PUNP6-9 -37 -46 -9 -148 -157
PUNP5-151 -9 -160 -40 -9 -49
PUNP5-9 -40 -49 -9 -151 -160
PUNP2-148 -10 -158 -38 -11 -49
PUNP2-11 -38 -49 -10 -148 -158
PUNP4-149 -11 -160 -37 -9 -46
PUNP4-9 -37 -46 -11 -149 -160
PUNP1-151 -10 -161 -42 -11 -53
PUNP1-11 -42 -53 -10 -151 -161
PUNP3-149 -11 -160 -38 -9 -47
PUNP3-9 -38 -47 -11 -149 -160
PUNP7-149 -9 -158 -38 -9 -47
PUNP7-9 -38 -47 -9 -149 -158
GUAD -539 -11 -550 -563 -38 -601
ADD -541 -11 -552 -562 -38 -600
L-LACD2 -6 -17 -23 -17 -57 -74
L-LACD3 -9 -19 -28 -13 -52 -65
ATPS4r-734 -729 -1463 -394 -572 -966
ATPS4r-572 -394 -966 -729 -734 -1463
CRNBTCT-2 -3 -5 -2 -3 -5
CRNBTCT-3 -2 -5 -3 -2 -5
CRNCBCT-3 -2 -5 -3 -2 -5
CRNCBCT-2 -3 -5 -2 -3 -5
CRNCDH-2 -234 -236 -2 -87 -89
CRNCDH-87 -2 -89 -234 -2 -236
CYTBD -467 -570 -1037 -347 -538 -885
CYTBO3 -467 -570 -1037 -347 -538 -885
LDH_D2 -6 -17 -23 -16 -57 -73
DMSOR1 -8 -241 -249 -8 -95 -103
DMSOR2 -3 -237 -240 -7 -94 -101
TMAOR1 -457 -241 -698 -332 -95 -427
TMAOR2 -452 -237 -689 -331 -94 -425
FDH2 -462 -335 -797 -342 -521 -863
FDH3 -465 -337 -802 -338 -516 -854
GLCDe -105 -330 -435 -249 -468 -717
G3PD7 -14 -11 -25 -12 -13 -25
G3PD6 -15 -12 -27 -16 -18 -34
G3PD5 -12 -10 -22 -20 -23 -43
HYD3 -454 -328 -782 -331 -453 -784
HYD1 -452 -327 -779 -339 -463 -802
HYD2 -455 -329 -784 -335 -458 -793
NO3R1 -463 -573 -1036 -329 -542 -871
NO3R2 -458 -569 -1027 -331 -545 -876
7
NADH5 -499 -53 -552 -375 -55 -430
NADH9 -501 -54 -555 -367 -45 -412
NADH10 -502 -55 -557 -371 -50 -421
NTRIR2x -500 -290 -790 -365 -161 -526
NADH6 -499 -377 -876 -375 -504 -879
NADH7 -502 -379 -881 -371 -499 -870
NADH8 -501 -378 -879 -367 -494 -861
THD2 -540 -417 -957 -380 -507 -887
POX -132 -14 -146 -260 -84 -344
SUCD4-10 -10 -20 -13 -15 -28
SUCD4-15 -13 -28 -10 -10 -20
TEST_NADTRHD -58 -56 -114 -93 -91 -184
TMAOR1e -457 -241 -698 -332 -95 -427
TMAOR2e -452 -237 -689 -331 -94 -425
DMSOR1e -8 -241 -249 -8 -95 -103
DMSOR2e -3 -237 -240 -7 -94 -101
TRDR -474 -26 -500 -367 -44 -411
PGL -86 -325 -411 -234 -450 -684
EDA -1 -23 -24 0 -55 -55
EDD -1 -233 -234 -1 -87 -88
GND -45 -54 -99 -19 -78 -97
RPE-3 -3 -6 -3 -4 -7
RPE-4 -3 -7 -3 -3 -6
RPI-8 -3 -11 -3 -3 -6
RPI-3 -3 -6 -3 -8 -11
TALA-13 -11 -24 -11 -11 -22
TALA-11 -11 -22 -11 -13 -24
TKT1-12 -11 -23 -6 -13 -19
TKT1-13 -6 -19 -11 -12 -23
TKT2-6 -17 -23 -6 -21 -27
TKT2-21 -6 -27 -17 -6 -23
G6PDH2r-52 -368 -420 -22 -466 -488
G6PDH2r-466 -22 -488 -368 -52 -420
GMPS2 -125 -367 -492 -418 -593 -1011
IMPD -130 -363 -493 -286 -497 -783
GMPR -471 -28 -499 -369 -81 -450
ADSS -11 -364 -375 -31 -603 -634
ADSL2r-1 -11 -12 -1 -12 -13
ADSL2r-12 -1 -13 -11 -1 -12
ADSL1r-1 -16 -17 0 -43 -43
ADSL1r-43 0 -43 -16 -1 -17
PRASCS-45 -395 -440 -190 -735 -925
PRASCS-735 -190 -925 -395 -45 -440
PRAGSr-43 -396 -439 -177 -735 -912
PRAGSr-735 -177 -912 -396 -43 -439
AIRC3-1 0 -1 -1 -1 -2
AIRC3-1 -1 -2 0 -1 -1
GLUPRT -88 -35 -123 -256 -107 -363
AICART-4 -5 -9 -4 -6 -10
AICART-6 -4 -10 -5 -4 -9
IMPC-89 -1 -90 -236 -1 -237
IMPC-1 -236 -237 -1 -89 -90
AIRC2 -38 -394 -432 -177 -735 -912
PRFGS -125 -422 -547 -418 -768 -1186
PRAIS -37 -395 -432 -172 -734 -906
GARFT-3 -329 -332 -5 -455 -460
GARFT-455 -5 -460 -329 -3 -332
GART -48 -395 -443 -179 -735 -914
ASPCT -9 -360 -369 -18 -596 -614
DHORTS-86 -324 -410 -235 -450 -685
DHORTS-450 -235 -685 -324 -86 -410
DHORD2 -3 -3 -6 -15 -16 -31
DHORD5 -6 -5 -11 -11 -11 -22
ORPT-72 -11 -83 -8 -3 -11
ORPT-3 -8 -11 -11 -72 -83
OMPDC -449 -9 -458 -325 -65 -390
CTPS2 -125 -428 -553 -423 -770 -1193
CBMK -129 -359 -488 -187 -590 -777
AMANAPE -1 0 -1 0 -1 -1
AMANK -37 -358 -395 -173 -589 -762
ALLTNt2r-450 -326 -776 -326 -450 -776
ALLTNt2r-450 -326 -776 -326 -450 -776
ARGORNt7-9 -12 -21 -12 -9 -21
ARGORNt7-9 -12 -21 -12 -9 -21
ACACt2-450 -326 -776 -326 -450 -776
ACACt2-450 -326 -776 -326 -450 -776
BUTt2r-450 -326 -776 -326 -450 -776
BUTt2r-450 -326 -776 -326 -450 -776
CYNTt2 -449 -325 -774 -325 -449 -774
GALCTt2r-450 -326 -776 -326 -450 -776
GALCTt2r-450 -326 -776 -326 -450 -776
GLCRt2r-450 -326 -776 -326 -450 -776
GLCRt2r-450 -326 -776 -326 -450 -776
PPPNt2r-450 -326 -776 -326 -450 -776
PPPNt2r-450 -326 -776 -326 -450 -776
HPPPNt2r-450 -326 -776 -326 -450 -776
HPPPNt2r-450 -326 -776 -326 -450 -776
HCINNMt2r-450 -326 -776 -326 -450 -776
HCINNMt2r-450 -326 -776 -326 -450 -776
GLUABUTt7-38 -31 -69 -31 -38 -69
GLUABUTt7-38 -31 -69 -31 -38 -69
ALAt2r-456 -334 -790 -334 -456 -790
ALAt2r-456 -334 -790 -334 -456 -790
URAt2r-454 -329 -783 -329 -454 -783
URAt2r-454 -329 -783 -329 -454 -783
GLYBt2r-453 -326 -779 -326 -453 -779
GLYBt2r-453 -326 -779 -326 -453 -779
CHLabc -126 -396 -522 -407 -737 -1144
GLYBabc -127 -396 -523 -407 -738 -1145
TARTRt7-22 -14 -36 -14 -22 -36
TARTRt7-22 -14 -36 -14 -22 -36
SUCCabc -144 -406 -550 -417 -755 -1172
GUAt2 -453 -330 -783 -330 -453 -783
XANt2 -453 -328 -781 -328 -453 -781
ACKr-49 -35 -84 -175 -140 -315
ACKr-140 -175 -315 -35 -49 -84
ACS -72 -29 -101 -196 -118 -314
ADHEr-509 -73 -582 -378 -66 -444
ADHEr-66 -378 -444 -73 -509 -582
LDH_D-44 -376 -420 -53 -539 -592
LDH_D-539 -53 -592 -376 -44 -420
FHL -459 -10 -469 -329 -58 -387
PTAr-159 -22 -181 -52 -24 -76
PTAr-24 -52 -76 -22 -159 -181
PFL -66 -21 -87 -35 -23 -58
SDPTA-17 -28 -45 -21 -35 -56
SDPTA-35 -21 -56 -28 -17 -45
ASAD-191 -368 -559 -56 -467 -523
ASAD-467 -56 -523 -368 -191 -559
DHDPS -44 -557 -601 -16 -535 -551
DHDPRy -466 -18 -484 -367 -44 -411
THDPS -88 -21 -109 -237 -24 -261
SDPDS -86 -12 -98 -234 -22 -256
DAPE-1 -3 -4 0 -1 -1
DAPE-1 0 -1 -3 -1 -4
LYSDC -453 -8 -461 -328 -59 -387
THRAr-5 -9 -14 -6 -10 -16
THRAr-10 -6 -16 -9 -5 -14
8
DAPDC -450 -12 -462 -327 -62 -389
HSDy-44 -369 -413 -20 -467 -487
HSDy-467 -20 -487 -369 -44 -413
ASPK-44 -35 -79 -189 -140 -329
ASPK-140 -189 -329 -35 -44 -79
HSK -37 -358 -395 -174 -588 -762
THRS -86 -42 -128 -233 -151 -384
12PPDt-2 -2 -4 -2 -2 -4
12PPDt-2 -2 -4 -2 -2 -4
NMNt7 -88 -327 -415 -237 -458 -695
ACALDt-5 -4 -9 -4 -5 -9
ACALDt-5 -4 -9 -4 -5 -9
GUAt-4 -6 -10 -6 -4 -10
GUAt-4 -6 -10 -6 -4 -10
HYXNt-4 -4 -8 -4 -4 -8
HYXNt-4 -4 -8 -4 -4 -8
XANt-4 -4 -8 -4 -4 -8
XANt-4 -4 -8 -4 -4 -8
NACUP -2 -1 -3 -1 -2 -3
ASNabc -127 -397 -524 -408 -738 -1146
ASNt2r-453 -327 -780 -327 -453 -780
ASNt2r-453 -327 -780 -327 -453 -780
DAPabc -124 -397 -521 -408 -735 -1143
CYSabc -125 -400 -525 -411 -736 -1147
ACt2r-461 -327 -788 -327 -461 -788
ACt2r-461 -327 -788 -327 -461 -788
ETOHt2r-451 -326 -777 -326 -451 -777
ETOHt2r-451 -326 -777 -326 -451 -777
PYRt2r-492 -338 -830 -338 -492 -830
PYRt2r-492 -338 -830 -338 -492 -830
O2t-4 -20 -24 -20 -4 -24
O2t-4 -20 -24 -20 -4 -24
CO2t-58 -8 -66 -8 -58 -66
CO2t-58 -8 -66 -8 -58 -66
H2Ot-86 -233 -319 -233 -86 -319
H2Ot-86 -233 -319 -233 -86 -319
DHAt-3 -3 -6 -3 -3 -6
DHAt-3 -3 -6 -3 -3 -6
NH3t-34 -7 -41 -7 -34 -41
NH3t-34 -7 -41 -7 -34 -41
ARBt2r-452 -327 -779 -327 -452 -779
ARBt2r-452 -327 -779 -327 -452 -779
ARBabc -126 -397 -523 -408 -737 -1145
HISt2r-452 -326 -778 -326 -452 -778
HISt2r-452 -326 -778 -326 -452 -778
PHEt2r-451 -326 -777 -326 -451 -777
PHEt2r-451 -326 -777 -326 -451 -777
LEUt2r-452 -326 -778 -326 -452 -778
LEUt2r-452 -326 -778 -326 -452 -778
VALt2r-453 -327 -780 -327 -453 -780
VALt2r-453 -327 -780 -327 -453 -780
ILEt2r-452 -327 -779 -327 -452 -779
ILEt2r-452 -327 -779 -327 -452 -779
CBL1abc -124 -395 -519 -406 -735 -1141
CADVt -454 -328 -782 -328 -454 -782
CRNt7 -3 -3 -6 -3 -3 -6
NAt3_1-457 -332 -789 -332 -457 -789
NAt3_1-457 -332 -789 -332 -457 -789
CITt7 -23 -14 -37 -14 -23 -37
CSNt2 -450 -325 -775 -325 -450 -775
MALTpts -4 -13 -17 -25 -43 -68
ACGApts -2 -14 -16 -20 -44 -64
DALAt2r-453 -329 -782 -329 -453 -782
DALAt2r-453 -329 -782 -329 -453 -782
DSERt2r-450 -326 -776 -326 -450 -776
DSERt2r-450 -326 -776 -326 -450 -776
GLYt2r-454 -329 -783 -329 -454 -783
GLYt2r-454 -329 -783 -329 -454 -783
SULabc -123 -395 -518 -406 -734 -1140
ASPt2_2 -456 -341 -797 -341 -456 -797
FUMt2_2 -459 -334 -793 -334 -459 -793
MALt2_2 -453 -331 -784 -331 -453 -784
SUCCt2_2 -470 -336 -806 -336 -470 -806
ASPt2_3 -456 -341 -797 -341 -456 -797
MALt2_3 -453 -331 -784 -331 -453 -784
SUCCt2_3 -470 -336 -806 -336 -470 -806
SUCCt2b -470 -336 -806 -336 -470 -806
FUMt2_3 -459 -334 -793 -334 -459 -793
SUCFUMt-31 -22 -53 -22 -31 -53
SUCFUMt-31 -22 -53 -22 -31 -53
GALCTNt2r-450 -326 -776 -326 -450 -776
GALCTNt2r-450 -326 -776 -326 -450 -776
GALURt2r-451 -326 -777 -326 -451 -777
GALURt2r-451 -326 -777 -326 -451 -777
GLCURt2r-451 -326 -777 -326 -451 -777
GLCURt2r-451 -326 -777 -326 -451 -777
OCDCAt2 -449 -325 -774 -325 -449 -774
HDCAt2 -455 -326 -781 -326 -455 -781
TTDCAt2 -455 -326 -781 -326 -455 -781
FE2abc -123 -396 -519 -407 -734 -1141
FORt-10 -5 -15 -5 -10 -15
FORt-10 -5 -15 -5 -10 -15
FUCt-451 -326 -777 -326 -451 -777
FUCt-451 -326 -777 -326 -451 -777
ABUTt2 -453 -327 -780 -327 -453 -780
GALt2 -453 -326 -779 -326 -453 -779
GLCt2 -465 -327 -792 -327 -465 -792
GALTpts -2 -14 -16 -20 -44 -64
GLNabc -124 -406 -530 -417 -735 -1152
GLYCt-4 -5 -9 -5 -4 -9
GLYCt-4 -5 -9 -5 -4 -9
GLYALDt-2 -2 -4 -2 -2 -4
GLYALDt-2 -2 -4 -2 -2 -4
UREAt-3 -1 -4 -1 -3 -4
UREAt-3 -1 -4 -1 -3 -4
GLYC3Pt6 -155 -43 -198 -43 -155 -198
ASPabc -130 -411 -541 -422 -741 -1163
GLUabc -157 -422 -579 -433 -768 -1201
ASPt2 -456 -341 -797 -341 -456 -797
GLUt2r-483 -352 -835 -352 -483 -835
GLUt2r-483 -352 -835 -352 -483 -835
GLUt4 -42 -36 -78 -36 -42 -78
ORNabc -129 -401 -530 -412 -740 -1152
ARGabc -126 -399 -525 -410 -737 -1147
HISabc -126 -396 -522 -407 -737 -1144
LYSabc -127 -398 -525 -409 -738 -1147
IDONt2r-454 -326 -780 -326 -454 -780
IDONt2r-454 -326 -780 -326 -454 -780
GLCNt2r-454 -327 -781 -327 -454 -781
GLCNt2r-454 -327 -781 -327 -454 -781
DDGLCNt2r-452 -326 -778 -326 -452 -778
DDGLCNt2r-452 -326 -778 -326 -452 -778
Kabc -125 -396 -521 -407 -736 -1143
AKGt2r-466 -344 -810 -344 -466 -810
AKGt2r-466 -344 -810 -344 -466 -810
LCTSt-450 -326 -776 -326 -450 -776
LCTSt-450 -326 -776 -326 -450 -776
ILEabc -126 -397 -523 -408 -737 -1145
9
THRabc -128 -400 -528 -411 -739 -1150
ALAabc -130 -404 -534 -415 -741 -1156
VALabc -127 -397 -524 -408 -738 -1146
LEUabc -126 -396 -522 -407 -737 -1144
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METDabc -123 -394 -517 -405 -734 -1139
GALabc -127 -396 -523 -407 -738 -1145
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ACNAMt2 -449 -325 -774 -325 -449 -774
NO3t7 -4 -5 -9 -5 -4 -9
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NO2t2r-453 -327 -780 -327 -453 -780
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PNTOt4 -9 -9 -18 -9 -9 -18
PIt2r-595 -360 -955 -360 -595 -955
PIt2r-595 -360 -955 -360 -595 -955
NMNP -2 -4 -6 -4 -2 -6
PTRCabc -129 -400 -529 -411 -740 -1151
SPMDabc -125 -397 -522 -408 -736 -1144
PTRCORNt7-12 -13 -25 -13 -12 -25
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PTRCt2r-455 -330 -785 -330 -455 -785
PROt2r-453 -328 -781 -328 -453 -781
PROt2r-453 -328 -781 -328 -453 -781
PROabc -127 -398 -525 -409 -738 -1147
PIabc -269 -394 -663 -441 -734 -1175
ACMANApts -3 -14 -17 -21 -44 -65
MNLpts -2 -14 -16 -20 -44 -64
SUCpts -2 -14 -16 -20 -43 -63
GLCpts -19 -18 -37 -23 -51 -74
FRUpts -3 -14 -17 -22 -43 -65
TREpts -3 -15 -18 -22 -44 -66
PROt4 -12 -12 -24 -12 -12 -24
RMNt -450 -325 -775 -325 -450 -775
TSULabc -123 -395 -518 -406 -734 -1140
SERt2r-456 -333 -789 -333 -456 -789
SERt2r-456 -333 -789 -333 -456 -789
THMabc -124 -395 -519 -406 -735 -1141
SBTpts -2 -14 -16 -20 -44 -64
SERt4 -15 -17 -32 -17 -15 -32
THRt4 -13 -14 -27 -14 -13 -27
TAURabc -123 -395 -518 -406 -734 -1140
THRt2r-454 -330 -784 -330 -454 -784
THRt2r-454 -330 -784 -330 -454 -784
TRPt2r-453 -326 -779 -326 -453 -779
TRPt2r-453 -326 -779 -326 -453 -779
Kt2r-451 -326 -777 -326 -451 -777
Kt2r-451 -326 -777 -326 -451 -777
TYRt2r-451 -327 -778 -327 -451 -778
TYRt2r-451 -327 -778 -327 -451 -778
GLYC3Pabc -132 -401 -533 -412 -743 -1155
MAN6Pt6_2 -149 -38 -187 -38 -149 -187
G6Pt6_2 -154 -40 -194 -40 -154 -194
FUCPt6_2 -146 -36 -182 -36 -146 -182
URAt2 -454 -329 -783 -329 -454 -783
XTSNt2r-452 -326 -778 -326 -452 -778
XTSNt2r-452 -326 -778 -326 -452 -778
INSt2r-454 -328 -782 -328 -454 -782
INSt2r-454 -328 -782 -328 -454 -782
ADNt2r-454 -330 -784 -330 -454 -784
ADNt2r-454 -330 -784 -330 -454 -784
CYTDt2r-452 -328 -780 -328 -452 -780
CYTDt2r-452 -328 -780 -328 -452 -780
THMDt2r-453 -328 -781 -328 -453 -781
THMDt2r-453 -328 -781 -328 -453 -781
URIt2r-454 -328 -782 -328 -454 -782
URIt2r-454 -328 -782 -328 -454 -782
XYLt2 -451 -326 -777 -326 -451 -777
XYLabc -125 -396 -521 -407 -736 -1143
CHLt2r-452 -326 -778 -326 -452 -778
CHLt2r-452 -326 -778 -326 -452 -778
ADEt2r-455 -329 -784 -329 -455 -784
ADEt2r-455 -329 -784 -329 -455 -784
RIBabc -123 -395 -518 -406 -734 -1140
PSCVT-4 -37 -41 -20 -146 -166
PSCVT-146 -20 -166 -37 -4 -41
DHQS 0 -36 -36 0 -147 -147
CHORS 0 -40 -40 -1 -146 -147
DHQD-1 -234 -235 0 -87 -87
DHQD-87 0 -87 -234 -1 -235
SHK3Dr-467 -19 -486 -368 -44 -412
SHK3Dr-44 -368 -412 -19 -467 -486
DDPA -91 -36 -127 -256 -146 -402
SHKK -37 -358 -395 -173 -589 -762
PPNDH -449 -241 -690 -325 -145 -470
CHORM 0 -1 -1 -4 0 -4
TRPAS2-90 -24 -114 -235 -80 -315
TRPAS2-80 -235 -315 -24 -90 -114
TRPS2 -10 -235 -245 -12 -90 -102
TRPS3 0 -12 -12 -1 -15 -16
TRPS1 -7 -244 -251 -10 -102 -112
PRAIi 0 0 0 0 0 0
IGPS -449 -242 -691 -324 -144 -468
ANS -1 -366 -367 -16 -526 -542
ANPRT -1 -7 -8 -11 -72 -83
PPND -41 -46 -87 -51 -106 -157
TYRTA-19 -28 -47 -23 -35 -58
10
TYRTA-35 -23 -58 -28 -19 -47
PHETA1-19 -28 -47 -22 -35 -57
PHETA1-35 -22 -57 -28 -19 -47
ATPM -123 -394 -517 -405 -734 -1139
BETALDHx -126 -364 -490 -284 -500 -784
BETALDHy -129 -369 -498 -252 -470 -722
HCO3E-144 -328 -472 -241 -450 -691
HCO3E-450 -241 -691 -328 -144 -472
CYANST 1 -323 -322 -1 -452 -453
CAT -4 -253 -257 -1 -90 -91
SELNPS -122 -41 -163 -405 -179 -584
SPODM -448 -21 -469 -324 -8 -332
ACONMT -1 0 -1 -7 -6 -13
THRD_L -5 -8 -13 -6 -34 -40
ACHBS -492 -8 -500 -339 -58 -397
ACLS -492 -8 -500 -338 -58 -396
KARA2i -466 -18 -484 -367 -44 -411
KARA1i -466 -18 -484 -367 -44 -411
DHAD2 0 -233 -233 0 -87 -87
DHAD1 0 -236 -236 0 -87 -87
ILETA-20 -28 -48 -23 -35 -58
ILETA-35 -23 -58 -28 -20 -48
VALTA-21 -31 -52 -23 -35 -58
VALTA-35 -23 -58 -31 -21 -52
LEUTAi -34 -22 -56 -28 -20 -48
IPPS -100 -345 -445 -252 -473 -725
IPMD -41 -362 -403 -51 -496 -547
OMCDC -449 -8 -457 -324 -58 -382
IPPMIa0 -234 -234 -1 -87 -88
IPPMIa-87 -1 -88 -234 0 -234
IPPMIb-87 0 -87 -234 -1 -235
IPPMIb-1 -234 -235 0 -87 -87
... For the directed hypergraph case, these curvatures were introduced recently and very little is known about their descriptive power. In this paper, we first present the results of our discretizations of Forman-Ricci [1] and Ollivier-Ricci [2] curvature notions, then, we show that they are powerful tools for exploring local properties of directed hypergraph motifs. To conclude, we carry out a curvature-based analysis of the metabolic network of E. coli. ...
... Edges connecting nodes with large degree have very negative Forman-Ricci curvature values, allowing a readily identification of those edges playing a key role in the cohesion of a network. We generalize this notion to directed hypergraphs, [1]. Formally, a directed hypergraph is a couple H = (V, E) where V is a set of vertices and E a set of ordered pairs of subsets of V called hyperedges. ...
... 90% of chemical reactions have at most three reactants and three products (also observed for the whole Chemical Space [3]), which, according to equation 1, indicates that frequent curvature values in Fig. 2b) are ruled by the accumulated in-and out-degree. In particular, frequent values of curvature were found to distinguish bottle neck and redundant reactions in the metabolic network [1]. On the other hand, when considering the number of incoming neighbors of reactants and of outgoing neighbors of products for every reaction, frequencies are of the order of hundreds and, for some reactions, almost the whole substrate set, as shown in Fig. 2c). ...
Conference Paper
Full-text available
Networks encoding symmetric binary relations between pairs of elements are mathematically represented by (undirected) graphs. Graph theory is a well developed mathematical subject, but empirical networks are typically less regular and also often much larger than the graphs that are mathematically best understood. Several quantities have therefore been introduced to characterize the large scale behavior or to identify the most important vertices in empirical networks. As the crucial structure of a graph is, however, given by the set of its edges rather than by its vertices, we should systematically define and evaluate quantities assigned to the edges rather than to the vertices. Curvature is a notion originally introduced in the context of smooth Riemannian manifolds to measure local or global deviation of a manifold from being Euclidean. Ricci curvature specifically, as a local measure, provides relatively broad information about the structure of positively curved manifolds. Therefore, there have been several attempts to discretize curvature notions to other settings such as cell complexes, graphs and undirected hypergraphs for obtaining similar results. By this discretizations they have been able to transfer some of the analytical or topological properties of original smooth curvatures to these discrete spaces. For the directed hypergraph case, these curvatures were introduced recently and very little is known about their descriptive power. In this paper, we first present the results of our discretizations of Forman-Ricci and Ollivier-Ricci curvature notions, then, we show that they are powerful tools for exploring local properties of directed hypergraph motifs. To conclude, we carry out a curvature-based analysis of the metabolic network of E. coli.
... More precisely, it has been discovered that concepts of curvature can be formulated in such a way that they apply naturally not only to smooth Riemannian manifolds, but also to various kinds of discrete spaces (Forman 2003;Saucan 2019), like graphs (Jost and Liu 2014;Ollivier 2007) or hypergraphs (Asoodeh et al. 2018;Banerjee 2020). Much effort has focused on concepts of Ricci curvature in this context, and that is also what we shall explore in this paper, drawing on recent theoretical work from our group, like notions of such Ricci curvature for directed hypergraphs (Eidi and Jost 2020;Leal et al. 2018). ...
... In this paper, after a short history of curvature notions, and in particular Ricci curvature, from the beginnings to some recent advances, we present the results of our generalizations of Forman-Ricci (Leal et al. 2018) and Ollivier-Ricci (Eidi and Jost 2020) curvature notions to directed hypergraphs. Then, we show that they are powerful tools for exploring local properties of directed hypergraph motifs. ...
... Here, we recall (Leal et al. 2018) where we have developed Forman-Ricci Curvature for directed hypergraphs. Formally, a directed hypergraph is a couple H = (V , E) where V is a set of vertices and E a set of ordered pairs of subsets of V called hyperedges. ...
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Relationships in real systems are often not binary, but of a higher order, and therefore cannot be faithfully modelled by graphs, but rather need hypergraphs. In this work, we systematically develop formal tools for analyzing the geometry and the dynamics of hypergraphs. In particular, we show that Ricci curvature concepts, inspired by the corresponding notions of Forman and Ollivier for graphs, are powerful tools for probing the local geometry of hypergraphs. In fact, these two curvature concepts complement each other in the identification of specific connectivity motifs. In order to have a baseline model with which we can compare empirical data, we introduce a random model to generate directed hypergraphs and study properties such as degree of nodes and edge curvature, using numerical simulations. We can then see how our notions of curvature can be used to identify connectivity patterns in the metabolic network of E. coli that clearly deviate from those of our random model. Specifically, by applying hypergraph shuffling to this metabolic network we show that the changes in the wiring of a hypergraph can be detected by Forman Ricci and Ollivier Ricci curvatures.
... They will be published on our blog which is accessible at https://blog.twitterexplorer.org. Furthermore, it is planned to add the possibility of exploring recently developed measures such as graph curvatures which can provide new insights to the analysis of social networks (Leal et al. 2018). The authors plan to actively maintain the tool and adapt it to Twitter API changes, like the one that was recently announced for Academic Research (Twitter 2021). ...
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We present an open-source interface for scientists to explore Twitter data through interactive network visualizations. Combining data collection, transformation and visualization in one easily accessible framework, the twitter explorer connects distant and close reading of Twitter data through the interactive exploration of interaction networks and semantic networks. By lowering the technological barriers of data-driven research, it aims to attract researchers from various disciplinary backgrounds and facilitates new perspectives in the thriving field of computational social science.
... Our results contribute to the undergoing generalisation of network theory to hypergraphs, where the traditional network description as a graph is being abstracted to that of hypergraphs as a mean to model complex relations among multiple entities [43,65]. We show that hypergraphs can be ordered and that the resulting structure has been at the core of chemistry for more than 150 years. ...
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For more than 150 years, the structure of the periodic system of the chemical elements has intensively motivated research in different areas of chemistry and physics. However, there is still no unified picture of what a periodic system is. Herein, based on the relations of order and similarity, we report a formal mathematical structure for the periodic system, which corresponds to an ordered hypergraph. It is shown that the current periodic system of chemical elements is an instance of the general structure. The definition is used to devise a tailored periodic system of polarizability of single covalent bonds, where order relationships are quantified within subsets of similar bonds and among these classes. The generalized periodic system allows envisioning periodic systems in other disciplines of science and humanities.
... Our results contribute to the undergoing generalisation of network theory to hypergraphs, where the traditional network description as a graph is being abstracted to that of hypergraphs as a mean to model complex relations among multiple entities [43,65]. We show that hypergraphs can be ordered and that the resulting structure has been at the core of chemistry for more than 150 years. ...
Preprint
Full-text available
For more than 150 years the structure of the periodic system of the chemical elements has intensively motivated research in different areas of chemistry and physics. However, there is still no unified picture of what a periodic system is. Herein, based on the relations of order and similarity, we report a formal mathematical structure for the periodic system, which corresponds to an ordered hypergraph. It is shown that the current periodic system of chemical elements is an instance of the general structure. The definition is used to devise a tailored periodic system of polarizability of single covalent bonds, where order relationships are quantified within subsets of similar bonds and among these classes. The generalised periodic system allows envisioning periodic systems in other disciplines of science and humanities.
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We have performed an empirical comparison of two distinct notions of discrete Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci curvature were developed based on different properties of the classical smooth notion, and thus, the two notions shed light on different aspects of network structure and behavior. Nevertheless, our extensive computational analysis in a wide range of both model and real-world networks shows that the two discretizations of Ricci curvature are highly correlated in many networks. Moreover, we show that if one considers the augmented Forman-Ricci curvature which also accounts for the two-dimensional simplicial complexes arising in graphs, the observed correlation between the two discretizations is even higher, especially, in real networks. Besides the potential theoretical implications of these observations, the close relationship between the two discretizations has practical implications whereby Forman-Ricci curvature can be employed in place of Ollivier-Ricci curvature for faster computation in larger real-world networks whenever coarse analysis suffices.
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In synthesis planning, the goal is to synthesize a target molecule from available starting materials, possibly optimizing costs such as price or environmental impact of the process. Current algorithmic approaches to synthesis planning are usually based on selecting a bond set and finding a single good plan among those induced by it. We demonstrate that synthesis planning can be phrased as a combinatorial optimization problem on hypergraphs by modeling individual synthesis plans as directed hyperpaths embedded in a hypergraph of reactions (HoR) representing the chemistry of interest. As a consequence, a polynomial time algorithm to find the K shortest hyperpaths can be used to compute the K best synthesis plans for a given target molecule. Having K good plans to choose from has many benefits: it makes the synthesis planning process much more robust when in later stages adding further chemical detail, it allows one to combine several notions of cost, and it provides a way to deal with imprecise yield estimates. A bond set gives rise to a HoR in a natural way. However, our modeling is not restricted to bond set based approaches-any set of known reactions and starting materials can be used to define a HoR. We also discuss classical quality measures for synthesis plans, such as overall yield and convergency, and demonstrate that convergency has a built-in inconsistency which could render its use in synthesis planning questionable. Decalin is used as an illustrative example of the use and implications of our results.
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The human brain forms functional networks on all spatial scales. Modern fMRI scanners allow for resolving functional brain data in high resolution, enabling the study of large-scale networks that relate to cognitive processes. The analysis of such networks forms a cornerstone of experimental neuroscience. Due to the immense size and complexity of the underlying data sets, efficient evaluation and visualization pose challenges for data analysis. In this study, we combine recent advances in experimental neuroscience and applied mathematics to perform a mathematical characterization of complex networks constructed from fMRI data. We use task-related edge densities [Lohmann et al., 2016] for constructing networks whose nodes represent voxels in the fMRI data and whose edges represent the task-related changes in synchronization between them. This construction captures the dynamic formation of patterns of neuronal activity and therefore efficiently represents the connectivity structure between brain regions. Using geometric methods that utilize Forman-Ricci curvature as an edge-based network characteristic [Weber et al., 2017], we perform a mathematical analysis of the resulting complex networks. We motivate the use of edge-based characteristics to evaluate the network structure with geometric methods. Our results identify important structural network features including long-range connections of high curvature acting as bridges between major network components. The geometric features link curvature to higher order network organization that could aid in understanding the connectivity and interplay of brain regions in cognitive processes.
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Hypergraph states, a generalization of graph states, constitute a large class of quantum states with intriguing non-local properties and have promising applications in quantum information science and technology. In this paper, we generalize the concept of hypergraph state to the qudit case, i.e., each vertex in the generalized hypergraph represents a $d$-level quantum system instead of a qubit. The generalized hypergraphs are named $d$-hypergraphs, and the corresponding quantum states are called qudit hypergraph states. It is shown that $d$-hypergraphs and $d$-level hypergraph states have a one-to-one correspondence. We prove that if one part of a $d$-hypergraph is connected with the other part, the corresponding subsystems are entangled. More generally, the structure of a $d$-hypergraph reveals the entanglement property of the corresponding quantum state. Bell non-locality, an important resource in fulfilling quantum information tasks, is also investigated. These states' responses to the generalized $Z$ ($X$) operations and $Z$ ($X$) measurements are studied.
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We have recently introduced Forman's discretization of Ricci curvature to the realm of complex networks. Forman curvature is an edge-based measure whose mathematical definition elegantly encapsulates the weights of nodes and edges in a complex network. In this contribution, we perform a comparative analysis of Forman curvature with other edge-based measures such as edge betweenness, embeddedness and dispersion in diverse model and real networks. We find that Forman curvature in comparison to embeddedness or dispersion is a better indicator of the importance of an edge for the large-scale connectivity of complex networks. Based on the definition of the Forman curvature of edges, there are two natural ways to define the Forman curvature of nodes in a network. In this contribution, we also examine these two possible definitions of Forman curvature of nodes in diverse model and real networks. Based on our empirical analysis, we find that in practice the unnormalized definition of the Forman curvature of nodes with the choice of combinatorial node weights is a better indicator of the importance of nodes in complex networks.
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Chemical research unveils the structure of chemical space, spanned by all chemical species, as documented in more than 200 y of scientific literature, now available in electronic databases. Very little is known, however, about the large-scale patterns of this exploration. Here we show, by analyzing millions of reac- tions stored in the Reaxys database, that chemists have reported new compounds in an exponential fashion from 1800 to 2015 with a stable 4.4% annual growth rate, in the long run nei- ther affected by World Wars nor affected by the introduction of new theories. Contrary to general belief, synthesis has been the means to provide new compounds since the early 19th cen- tury, well before Wöhler’s synthesis of urea. The exploration of chemical space has followed three statistically distinguishable regimes. The first one included uncertain year-to-year output of organic and inorganic compounds and ended about 1860, when structural theory gave way to a century of more regular and guided production, the organic regime. The current organometal- lic regime is the most regular one. Analyzing the details of the synthesis process, we found that chemists have had preferences in the selection of substrates and we identified the workings of such a selection. Regarding reaction products, the discovery of new compounds has been dominated by very few elemental com- positions. We anticipate that the present work serves as a starting point for more sophisticated and detailed studies of the history of chemistry.
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The genetic material of a eukaryotic cell (one whose nucleus and other organelles, including mitochondria, are enclosed within membranes) comprises both nuclear DNA (ncDNA) and mitochondrial DNA (mtDNA). These differ markedly in several aspects but nevertheless must encode proteins that are compatible with one another for the proper functioning of the organism. Here, we introduce a network model of the hypothetical coevolution of the two most common modes of cellular division for reproduction: by mitosis (supporting asexual reproduction) and by meiosis (supporting sexual reproduction). Our model is based on a random hypergraph, with two nodes for each possible genotype, each encompassing both ncDNA and mtDNA. One of the nodes is necessarily generated by mitosis occurring at a parent genotype, the other by meiosis occurring at two parent genotypes. A genotype's fitness depends on the compatibility of its ncDNA and mtDNA. The model has two probability parameters, p and r, the former accounting for the diversification of ncDNA during meiosis, the latter for the diversification of mtDNA accompanying both meiosis and mitosis. Another parameter, λ, is used to regulate the relative rate at which mitosis- and meiosis-generated genotypes are produced. We have found that, even though p and r do affect the existence of evolutionary pathways in the network, the crucial parameter regulating the coexistence of the two modes of cellular division is λ. Depending on genotype size, λ can be valued so that either mode of cellular division prevails. Our study is closely related to a recent hypothesis that brings mitochondria to the center stage and views the appearance of cellular division by meiosis, as opposed to division by mitosis, as an evolutionary strategy for boosting ncDNA diversification to keep up with that of mtDNA. Our results indicate that this may well have been the case, thus lending support to the first hypothesis in the field to take into account the role of such ubiquitous and essential organelles as mitochondria.
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We describe an approach to the analysis of chemical (and other) networks that, in contrast to other schemes, is based on edges rather than vertices, naturally works with directed and weighted edges, extends to higher dimensional structures like simplicial complexes or hypergraphs, and can draw upon a rich body of theoretical insight from geometry. As the approach is motivated by Riemannian geometry, the crucial quantity that we work with is called Ricci curvature, although in the present setting, it is of course not a curvature in the ordinary sense, but rather quantifies the divergence properties of edges. In order to illustrate the method and its potential, we apply it to metabolic and gene co-expression networks and detect some new general features in such networks.
Chapter
Curvature is a concept originally developed in differential and Riemannian geometry. There are various established notions of curvature, in particular sectional and Ricci curvature. An important theme in Riemannian geometry has been to explore the geometric and topological consequences of bounds on those curvatures, like divergence or convergence of geodesics, convexity properties of distance functions, growth of the volume of distance balls, transportation distance between such balls, vanishing theorems for Betti numbers, bounds for the eigenvalues of the Laplace operator or control of harmonic functions. Several of these geometric properties turn out to be equivalent to the corresponding curvature bounds in the context of Riemannian geometry. Since those properties often are also meaningful in the more general framework of metric geometry, in recent years, there have been several research projects that turned those properties into axiomatic definitions of curvature bounds in metric geometry. In this contribution, after developing the Riemannian geometric background, we explore some of these axiomatic approaches. In particular, we shall describe the insights in graph theory and network analysis following from the corresponding axiomatic curvature definitions.