576 research outputs found
Spectral Analysis of Protein-Protein Interactions in Drosophila melanogaster
Within a case study on the protein-protein interaction network (PIN) of
Drosophila melanogaster we investigate the relation between the network's
spectral properties and its structural features such as the prevalence of
specific subgraphs or duplicate nodes as a result of its evolutionary history.
The discrete part of the spectral density shows fingerprints of the PIN's
topological features including a preference for loop structures. Duplicate
nodes are another prominent feature of PINs and we discuss their representation
in the PIN's spectrum as well as their biological implications.Comment: 9 pages RevTeX including 8 figure
On the push&pull protocol for rumour spreading
The asynchronous push&pull protocol, a randomized distributed algorithm for
spreading a rumour in a graph , works as follows. Independent Poisson clocks
of rate 1 are associated with the vertices of . Initially, one vertex of
knows the rumour. Whenever the clock of a vertex rings, it calls a random
neighbour : if knows the rumour and does not, then tells the
rumour (a push operation), and if does not know the rumour and knows
it, tells the rumour (a pull operation). The average spread time of
is the expected time it takes for all vertices to know the rumour, and the
guaranteed spread time of is the smallest time such that with
probability at least , after time all vertices know the rumour. The
synchronous variant of this protocol, in which each clock rings precisely at
times , has been studied extensively. We prove the following results
for any -vertex graph: In either version, the average spread time is at most
linear even if only the pull operation is used, and the guaranteed spread time
is within a logarithmic factor of the average spread time, so it is . In the asynchronous version, both the average and guaranteed spread times
are . We give examples of graphs illustrating that these bounds
are best possible up to constant factors. We also prove theoretical
relationships between the guaranteed spread times in the two versions. Firstly,
in all graphs the guaranteed spread time in the asynchronous version is within
an factor of that in the synchronous version, and this is tight.
Next, we find examples of graphs whose asynchronous spread times are
logarithmic, but the synchronous versions are polynomially large. Finally, we
show for any graph that the ratio of the synchronous spread time to the
asynchronous spread time is .Comment: 25 page
Achlioptas processes are not always self-averaging
We consider a class of percolation models, called Achlioptas processes,
discussed in [Science 323, 1453 (2009)] and [Science 333, 322 (2011)]. For
these the evolution of the order parameter (the rescaled size of the largest
connected component) has been the main focus of research in recent years. We
show that, in striking contrast to `classical' models, self-averaging is not a
universal feature of these new percolation models: there are natural Achlioptas
processes whose order parameter has random fluctuations that do not disappear
in the thermodynamic limit.Comment: 4 pages, 3 figures. Revised and expanded. Title change
Random graph model with power-law distributed triangle subgraphs
Clustering is well-known to play a prominent role in the description and
understanding of complex networks, and a large spectrum of tools and ideas have
been introduced to this end. In particular, it has been recognized that the
abundance of small subgraphs is important. Here, we study the arrangement of
triangles in a model for scale-free random graphs and determine the asymptotic
behavior of the clustering coefficient, the average number of triangles, as
well as the number of triangles attached to the vertex of maximum degree. We
prove that triangles are power-law distributed among vertices and characterized
by both vertex and edge coagulation when the degree exponent satisfies
; furthermore, a finite density of triangles appears as
.Comment: 4 pages, 2 figure; v2: major conceptual change
A preferential attachment model with random initial degrees
In this paper, a random graph process is studied and its
degree sequence is analyzed. Let be an i.i.d. sequence. The
graph process is defined so that, at each integer time , a new vertex, with
edges attached to it, is added to the graph. The new edges added at time
t are then preferentially connected to older vertices, i.e., conditionally on
, the probability that a given edge is connected to vertex i is
proportional to , where is the degree of vertex
at time , independently of the other edges. The main result is that the
asymptotical degree sequence for this process is a power law with exponent
, where is the power-law exponent
of the initial degrees and the exponent predicted
by pure preferential attachment. This result extends previous work by Cooper
and Frieze, which is surveyed.Comment: In the published form of the paper, the proof of Proposition 2.1 is
incomplete. This version contains the complete proo
Laplacian spectra of complex networks and random walks on them: Are scale-free architectures really important?
We study the Laplacian operator of an uncorrelated random network and, as an
application, consider hopping processes (diffusion, random walks, signal
propagation, etc.) on networks. We develop a strict approach to these problems.
We derive an exact closed set of integral equations, which provide the averages
of the Laplacian operator's resolvent. This enables us to describe the
propagation of a signal and random walks on the network. We show that the
determining parameter in this problem is the minimum degree of vertices
in the network and that the high-degree part of the degree distribution is not
that essential. The position of the lower edge of the Laplacian spectrum
appears to be the same as in the regular Bethe lattice with the
coordination number . Namely, if , and
if . In both these cases the density of eigenvalues
as , but the limiting behaviors near
are very different. In terms of a distance from a starting vertex,
the hopping propagator is a steady moving Gaussian, broadening with time. This
picture qualitatively coincides with that for a regular Bethe lattice. Our
analytical results include the spectral density near
and the long-time asymptotics of the autocorrelator and the
propagator.Comment: 25 pages, 4 figure
Models and Algorithms for Graph Watermarking
We introduce models and algorithmic foundations for graph watermarking. Our
frameworks include security definitions and proofs, as well as
characterizations when graph watermarking is algorithmically feasible, in spite
of the fact that the general problem is NP-complete by simple reductions from
the subgraph isomorphism or graph edit distance problems. In the digital
watermarking of many types of files, an implicit step in the recovery of a
watermark is the mapping of individual pieces of data, such as image pixels or
movie frames, from one object to another. In graphs, this step corresponds to
approximately matching vertices of one graph to another based on graph
invariants such as vertex degree. Our approach is based on characterizing the
feasibility of graph watermarking in terms of keygen, marking, and
identification functions defined over graph families with known distributions.
We demonstrate the strength of this approach with exemplary watermarking
schemes for two random graph models, the classic Erd\H{o}s-R\'{e}nyi model and
a random power-law graph model, both of which are used to model real-world
networks
Clustering in Complex Directed Networks
Many empirical networks display an inherent tendency to cluster, i.e. to form
circles of connected nodes. This feature is typically measured by the
clustering coefficient (CC). The CC, originally introduced for binary,
undirected graphs, has been recently generalized to weighted, undirected
networks. Here we extend the CC to the case of (binary and weighted) directed
networks and we compute its expected value for random graphs. We distinguish
between CCs that count all directed triangles in the graph (independently of
the direction of their edges) and CCs that only consider particular types of
directed triangles (e.g., cycles). The main concepts are illustrated by
employing empirical data on world-trade flows
Slow relaxation in the Ising model on a small-world network with strong long-range interactions
We consider the Ising model on a small-world network, where the long-range
interaction strength is in general different from the local interaction
strength , and examine its relaxation behaviors as well as phase
transitions. As is raised from zero, the critical temperature also
increases, manifesting contributions of long-range interactions to ordering.
However, it becomes saturated eventually at large values of and the
system is found to display very slow relaxation, revealing that ordering
dynamics is inhibited rather than facilitated by strong long-range
interactions. To circumvent this problem, we propose a modified updating
algorithm in Monte Carlo simulations, assisting the system to reach equilibrium
quickly.Comment: 5 pages, 5 figure
Properties of contact matrices induced by pairwise interactions in proteins
The total conformational energy is assumed to consist of pairwise interaction
energies between atoms or residues, each of which is expressed as a product of
a conformation-dependent function (an element of a contact matrix, C-matrix)
and a sequence-dependent energy parameter (an element of a contact energy
matrix, E-matrix). Such pairwise interactions in proteins force native
C-matrices to be in a relationship as if the interactions are a Go-like
potential [N. Go, Annu. Rev. Biophys. Bioeng. 12. 183 (1983)] for the native
C-matrix, because the lowest bound of the total energy function is equal to the
total energy of the native conformation interacting in a Go-like pairwise
potential. This relationship between C- and E-matrices corresponds to (a) a
parallel relationship between the eigenvectors of the C- and E-matrices and a
linear relationship between their eigenvalues, and (b) a parallel relationship
between a contact number vector and the principal eigenvectors of the C- and
E-matrices; the E-matrix is expanded in a series of eigenspaces with an
additional constant term, which corresponds to a threshold of contact energy
that approximately separates native contacts from non-native ones. These
relationships are confirmed in 182 representatives from each family of the SCOP
database by examining inner products between the principal eigenvector of the
C-matrix, that of the E-matrix evaluated with a statistical contact potential,
and a contact number vector. In addition, the spectral representation of C- and
E-matrices reveals that pairwise residue-residue interactions, which depends
only on the types of interacting amino acids but not on other residues in a
protein, are insufficient and other interactions including residue
connectivities and steric hindrance are needed to make native structures the
unique lowest energy conformations.Comment: Errata in DOI:10.1103/PhysRevE.77.051910 has been corrected in the
present versio
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