4,937 research outputs found
Reconciling long-term cultural diversity and short-term collective social behavior
An outstanding open problem is whether collective social phenomena occurring
over short timescales can systematically reduce cultural heterogeneity in the
long run, and whether offline and online human interactions contribute
differently to the process. Theoretical models suggest that short-term
collective behavior and long-term cultural diversity are mutually excluding,
since they require very different levels of social influence. The latter
jointly depends on two factors: the topology of the underlying social network
and the overlap between individuals in multidimensional cultural space.
However, while the empirical properties of social networks are well understood,
little is known about the large-scale organization of real societies in
cultural space, so that random input specifications are necessarily used in
models. Here we use a large dataset to perform a high-dimensional analysis of
the scientific beliefs of thousands of Europeans. We find that inter-opinion
correlations determine a nontrivial ultrametric hierarchy of individuals in
cultural space, a result unaccessible to one-dimensional analyses and in
striking contrast with random assumptions. When empirical data are used as
inputs in models, we find that ultrametricity has strong and counterintuitive
effects, especially in the extreme case of long-range online-like interactions
bypassing social ties. On short time-scales, it strongly facilitates a
symmetry-breaking phase transition triggering coordinated social behavior. On
long time-scales, it severely suppresses cultural convergence by restricting it
within disjoint groups. We therefore find that, remarkably, the empirical
distribution of individuals in cultural space appears to optimize the
coexistence of short-term collective behavior and long-term cultural diversity,
which can be realized simultaneously for the same moderate level of mutual
influence
Next nearest neighbour Ising models on random graphs
This paper develops results for the next nearest neighbour Ising model on
random graphs. Besides being an essential ingredient in classic models for
frustrated systems, second neighbour interactions interactions arise naturally
in several applications such as the colour diversity problem and graphical
games. We demonstrate ensembles of random graphs, including regular
connectivity graphs, that have a periodic variation of free energy, with either
the ratio of nearest to next nearest couplings, or the mean number of nearest
neighbours. When the coupling ratio is integer paramagnetic phases can be found
at zero temperature. This is shown to be related to the locked or unlocked
nature of the interactions. For anti-ferromagnetic couplings, spin glass phases
are demonstrated at low temperature. The interaction structure is formulated as
a factor graph, the solution on a tree is developed. The replica symmetric and
energetic one-step replica symmetry breaking solution is developed using the
cavity method. We calculate within these frameworks the phase diagram and
demonstrate the existence of dynamical transitions at zero temperature for
cases of anti-ferromagnetic coupling on regular and inhomogeneous random
graphs.Comment: 55 pages, 15 figures, version 2 with minor revisions, to be published
J. Stat. Mec
Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching
We present a deterministic distributed algorithm that computes a
-edge-coloring, or even list-edge-coloring, in any -node graph
with maximum degree , in rounds. This answers
one of the long-standing open questions of \emph{distributed graph algorithms}
from the late 1980s, which asked for a polylogarithmic-time algorithm. See,
e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and
Elkin. The previous best round complexities were by
Panconesi and Srinivasan [STOC'92] and
by Fraigniaud, Heinrich, and Kosowski [FOCS'16]. A corollary of our
deterministic list-edge-coloring also improves the randomized complexity of
-edge-coloring to poly rounds.
The key technical ingredient is a deterministic distributed algorithm for
\emph{hypergraph maximal matching}, which we believe will be of interest beyond
this result. In any hypergraph of rank --- where each hyperedge has at most
vertices --- with nodes and maximum degree , this algorithm
computes a maximal matching in rounds.
This hypergraph matching algorithm and its extensions lead to a number of
other results. In particular, a polylogarithmic-time deterministic distributed
maximal independent set algorithm for graphs with bounded neighborhood
independence, hence answering Open Problem 5 of Barenboim and Elkin's book, a
-round deterministic
algorithm for -approximation of maximum matching, and a
quasi-polylogarithmic-time deterministic distributed algorithm for orienting
-arboricity graphs with out-degree at most ,
for any constant , hence partially answering Open Problem 10 of
Barenboim and Elkin's book
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